empirical statistical mechanics · 2008-09-25 · empirical statistical mechanics wang, xiaoming...
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Empirical Statistical Mechanics
Wang Xiaomingwxmmathfsuedu
Florida State University
Stochastic and Probabilistic Methods in Ocean-AtmosphereDynamics Victoria July 2008
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
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minus20 minus10 0 10 200
2000
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8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)
Least action principle
A =
int t
t0L(x(s) x(s) s) ds
Lagrange equations of motion
partLpartxj
minus ddtpartLpartxj
= 0
HamiltonianH =
sumj
pj xj minus L pj =partLpartxj
Hamiltonian system (common notation qj = xj )
dxj
dt=partHpartpj
dpj
dt= minuspartH
partxj
Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 2 Logistic map
T (x) = 4x(1minus x) x isin [01]
Figure Sensitive dependence
0 20 40 60 80 100 120 140 160 180 2000
01
02
03
04
05
06
07
08
09
1
Orbits of the Logistic map with close ICs
Figure Statistical coherence
-02 0 02 04 06 08 1 120
100
200
300
400
500
600
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 3 Lorenz 96 model
Edward Norton Lorenz 1917-2008
Lorenz96 duj
dt= (uj+1 minus ujminus2)ujminus1 minus uj + F
j = 01 middot middot middot J J = 5F = minus12
Figure Sensitive dependence
(Sensitive dependence)
Figure Statistical coherence
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
minus20 minus10 0 10 200
2000
4000
6000
8000
10000
12000
14000
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example 4 Rayleigh-Beacutenard convection
1Pr
(partupartt
+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0
partθ
partt+ u middot nablaθ minus u3 = ∆θ θ|z=01 = 0 (θ = T minus (1minus z))
Pr =ν
κ
Ra =gα(Tbottom minus Ttop)h3
νκ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Figure Lord Rayleigh (JohnWilliam Strutt) 1842-1919
Figure Henri Beacutenard (left)1874-1939
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
RBC set-up and numerics
Figure RBC set-up Figure Numerical simulation
(infin Pr simulation)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Figure Lord Rayleigh (JohnWilliam Strutt) 1842-1919
Figure Henri Beacutenard (left)1874-1939
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
RBC set-up and numerics
Figure RBC set-up Figure Numerical simulation
(infin Pr simulation)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
RBC set-up and numerics
Figure RBC set-up Figure Numerical simulation
(infin Pr simulation)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistical Approaches
dudt
= F(u) u isin H
S(t) solutionsemigroup
Long time average
lt Φ gt= limTrarrinfin
1T
int T
0Φ(v(t)) dt
Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages
lt Φ gtt=
intH
Φ(v) dmicrot(v)
microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N
lt Φ gtt=1N
Nsumj=1
Φ(vj(t))
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Statistics Solutions
microt t ge 0 dvdt
= F(v) S(t) t ge 0
Pull-backmicro0(Sminus1(t)(E)) = microt(E)
Push-forward (Φ suitable test functional)intH
Φ(v) dmicrot(v) =
intH
Φ(S(t)v) dmicro0(v)
Finite ensemble example
micro0 =Nsum
j=1
pjδv0j (v) microt =Nsum
j=1
pjδvj (t)(v)vj(t) = S(t)v0j
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Relative stability of statistical properties
Finite ensemble1N
Nsumj=1
Φ(uj(t))
Total variation
|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Liouvillersquos equations
Figure Joseph Liouville1809-1882
Liouville type equation
ddt
intH
Φ(v) dmicrot(v)
=
intHlt Φprime(v)F(v) gt dmicrot(v)
Φ good test functionals eg
Φ(v) = φ((vv1) middot middot middot (vvN))
Liouville equation (finite d)
part
parttp(v t) +nabla middot (p(v t)F(v)) = 0
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Hopfrsquos equations
Figure Eberhard Hopf1902-1983
Hopfrsquos equation (special case of Li-ouville type)
ddt
intH
ei(vg) dmicrot(v)
=
intH
i lt F(v)g gt ei(vg) dmicrot(v)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Invariant measures (IM)(Stationary StatisticsSolutions)
Invariant measure (IM) micro isin PM(H)
micro(Sminus1(t)(E) = micro(E)
Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Birkhoffrsquos Ergodic Theorem
Figure George DavidBirkhoff 1884-1944 Definition
micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E
Theorem (Birkhoffrsquos ErgodicTheorem)
If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie
limTrarrinfin
1T
int T
0ϕ(S(t)u) dt =
intHϕ(u) dmicro(u) as
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 1 dissipative
drdt
= r(1minus r2)dθdt
= 1
micro1 = δ0
dmicro2 =1
2πdθ on r = 1
Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Examples of IM 2 Hamiltonian
Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution
dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =
intH
exp(minusβH(pq)) dpdq
Z partition function β inverse temperatureMicro-canonical measuredistribution
dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq
χ =
intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(
Sk
)
χ structure function k Boltzmann const
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Shannonrsquos entropy (measuring uncertainty)
Figure Claude Shannon1916-2001
Definition (Shannon entropy)
S(p) = S(p1 pn) = minusnsum
i=1
pi ln pi
p isin PMn(A) probability measureon A = a1 an
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Uniqueness of Shannon entropy
Figure Edwin ThompsonJaynes 1922-1998
Theorem (Jaynes)
Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1
⋃A2
A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn
Hn(p1 pn) = H2(w1w2)
+w1Hk (p1w1 pkw1)
+w2Hnminusk (pk+1w2 pnw2)
Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK
sumnj=1 pj ln pj
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1
Fj =lt fj gtp=nsum
i=1
fj(ai)pi j = 1 r
or continuous version
Fj =lt fj gtp=
intfj(x)p(x) dx
Definition (Empirical maximum entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(p) = S(plowast) plowast isin C
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
plowasti =exp
(minus
sumrj=1 λj fj(ai)
)Z
Z =nsum
i=1
exp
minus rsumj=1
λj fj(ai)
Z partition function
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum relative entropy principle
Relative entropy
S(pp0) = minusNsum
j=1
pj lnpj
p0j
= minusP(pp0)
with prior distribution p0
Relative entropy as semi-distance
minusS(pp0) ge 0forallp
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pp0) = S(plowastp0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example discrete case
No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy principle for continuous pdf
Entropy
S(p) = minusint
p(x) ln p(x) dx
Relative entropy
S(pΠ0) = minusint
p(x) lnp(x)
Π0(x)dx = minusP(pΠ0)
with prior distribution Π0
Definition (Empirical maximum relative entropy principle)
Least biased most probable pdf plowast is given by
maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C
for a given set of constraints C defined by
C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Example continuous case
The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)
1Z
exp(minusβH)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω
(application to PDE)
ρ(x λ) ge 0intρ(x λ) dλ = 1ae
int q+
qminusρ(x λ) dλ = Probqminus le q(x) lt q+
lt F (q) gtρ=1|Ω|
intΩ
intR1
F (x λ)ρ(x λ) dλdx
S(ρ) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln ρ(x λ) dλdx
S(ρΠ0) = minus 1|Ω|
intΩ
intR1ρ(x λ) ln
ρ(x λ)
Π0(x λ)dλdx
Maximum entropy principle remain the same
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Barotropic quasi-geostrophic equation with topography
Barotropic quasi-geostrophic equation
partqpartt
+nablaperpψ middot nablaq = 0 q = ∆ψ + h
q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E
E = minus 12|Ω|
intΩ
ψ∆ψ dx
E =1
2|Ω|
intΩ
q2 dx
There exist infinitely many conserved quantities
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG
Energy and total enstrophy as constraints
E(ρ) = minus 12|Ω|
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
E(ρ) =12
int intR1λ2ρ(x λ) dλdx
Lagrangian multiplier calculation
δSδρ
= minus1minus ln ρ
δEδρ
=12λ2
δEδρ
= λψ(x)δqδρ
= λ
δintρ(x λ) dλδρ
= δx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum entropy principle to barotropicQG continued
Most probable state ρlowast
minus1minus ln(ρlowast) = minusθλψlowast +α
2λ2 + γ(x)
ρlowast(x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Application of maximum relative entropy principle tobarotropic QG
Energy constraint
E(ρ) = minus12
intΩ
ψ(q minus h) q(x) =
intR1λρ(x λ)dλ q = ∆ψ + h
Gaussian prior
Π0(λ) =
radicαradic2π
exp(minusα2λ2)
Most probable state
ρlowast(~x λ) =
radicαradic2π
exp(minusα2
(λminus θ
αψlowast)2)
θ Lagrange multiplier for energyMean field equation
∆ψlowast + h =θ
αψlowast = microψlowastwith micro =
θ
α α gt 0
Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Energy circulation theory with a general prior
BQG with bottom topography and channel geometryΩ = (02π)times (0 π)
ψ = ψprime =sumkge1
sumjge0
(ajk cos(jx) + bjk sin(jx)) sin(ky)
Energy and circulation as conserved quantities
E(ρ) = minus 12|Ω|
intΩ
ψωdx Γ(ρ) =1|Ω|
intΩ
intλρ(x λ)dλdx
Most probable one-point statistics
ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int
e(θψlowastminusγ)λΠ0(x λ) dλ
θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation
∆ψlowast+h =1θ
(part
partψlnZ(θψ x)
) ∣∣∣∣ψ=ψlowast
Z(ψx) =
inte(ψminusγ)λΠ0(x λ) dλ
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Mean field statistical theory for point vortices
2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1
|Ω|intΩ|x|2ω) as
conserved quantitiesAppropriate prior distribution
Nsumi=1
1N
(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|
= Π0(λ)
Mean field equation for the most probable state
∆ψlowast = ωlowast =|Ω|
intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int
e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx
Case with no angular momentum
∆ψlowast =Γ0eminusβψ
lowastinteminusβψlowastdx
β = minusθω0
Case with angular momentum
∆ψlowast =Γ0eminusβψ
lowastminusα|x|2intBR(0)
eminusβψlowastminusα|x|2dx
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics
Summary empirical statistical mechanics
Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation
q = G(ψ)
Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006
Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics