Н-theorem and entropy over boltzmann and poincare vedenyapin v.v., adzhiev s.z
TRANSCRIPT
Н-THEOREM and ENTROPY
over BOLTZMANN and POINCARE
Vedenyapin V.V.,Adzhiev S.Z.
Н-THEOREM and ENTROPY over BOLTZMANN and POINCARE
1.Boltzmann equation (Maxwell, 1866). H-theorem (Boltzmann,1872). Maxwell (1831-1879) and Boltzmann (1844-1906).2.Generalized versions of Boltzmann equation and its discrete models. H-theorem for chemical classical and quantum kinetics.
3.H.Poincare-V.Kozlov-D.Treschev version of H-theorem for Liouville equations.
The discrete velocity models of the Boltzmann equation and
of the quantum kinetic equations
We consider the Н-theorem for such generalization of equations of chemical kinetics, which involves the discrete velocity models of the quantum kinetic equations.
is a distribution function of particles in space point x at a time t, with mass and momentum , if is an average number of particles in one quantum state, because the number of states in is
models the collision integral.
for fermions, for bosons, for the Boltzmann (classical) gas:
nii
i
ii ffFf
mt
f,,, 1
x
p
3, htfi x
ni ,,2,1
im ip
jlk
lkjijilkijklni ffffffffffF
,,1 1111,,
11 0
jlkjilk
ijklni ffffffF
,,1 ,,
x,tfixp 3hxp
The Carleman model
,
,
22
21
2
21
22
1
ffdt
df
ffdt
df
1ln1ln 2211 ffffSH ff
0lnln 21
2221
21
22
21
fffffff
H
f
H
dt
dH fff
0 yx eexy
constAff 21
021
dt
ffd
AffHL 21, fλf
0f
f
λL ,0
0,0
λ
λL f 0
,0
λ
λL f
The Carleman model and its generalizations
,2exp2exp
,2exp2exp
2
21
1
12
2
1
12
2
21
1
f
GK
f
GK
dt
df
f
GK
f
GK
dt
df
fff
fff
,
,
22
21
2
21
22
1
ffdt
df
ffdt
df
.11
,11
21
22
22
21
2
22
21
21
22
1
ffffdt
df
ffffdt
df
1
12
2
21 2exp2exp
f
GK
f
GK
ξξ
The Н-theoremfor generalization of the Carleman model
0 yx eexy
ξff GGH
1
12
2
21 2exp2exp
f
GK
f
GK
ξξ
1
12
2
21
21
2exp2expf
GK
f
GK
f
H
f
H
dt
dH fff
fff
2
21
2211
2expf
GK
f
G
f
G
f
G
f
G ξf
ξfξf
02exp2exp
1122
f
G
f
G
f
G
f
G ξfξf
fξff ,GGH
The Markoff process (the random walk)with two states and its generalizations
,
,
22
1112
2
1122
21
1
fKfKdt
df
fKfKdt
df
,11
,11
122
12112
2
211212
21
1
ffKffKdt
df
ffKffKdt
df
j
mmjj
jmjm
m hKhKffdt
df 11
nm ,,1 mmm ffh 1
Equations of chemical kinetics
βα
ααβf
,ii
i Kdt
df ni ,,2,1
n21 αααn21 fff αf
n ,,, 21 α
n ,,, 21 β
nn
K
nn SSSSSS 22112211
αβ
βα,
CEDS
βα
αβ KK
β
βα
β
ααβ
βξξ KK
n
i i
ii
ffHS
1
1ln
ff
ββα
ααβ KK
βα
βff βα
ααβ
,i
i KKdt
df
Н-theorem for generalization of equations of chemical kinetics
The generalization of the principle of detailed balance:
Let the system is solved for initial data from M, whereis defined and continuous.Let M is strictly convex, and G is strictly convex on M.
ff βα
αβ
ξβξα βα
αβ
GG ee KK ,, ~~
ni ,,2,1
βα
αβ
αβ
fαf
,ii
i GeKdt
df ,~
G
βα
βα
αβ
αβ
fβfαf
,i
i GG ee KKdt
df ,, ~~
The statement of the theorem
Let the coefficients of the system are such that there exists at least one solution in M of generalization of the principle of detailed balance:
Then:
a) H-function does not increase on the solutions of the system. All stationary solutions of the system satisfy the generalization of detailed balance;
b) the system has n-r conservation laws of the form , where r is the dimension of the linear span of vectors , and vectors orthogonal to all . Stationary solution is unique, if we fix all the constants of these conservation laws, and is given by formula
where the values are determined by ; c) such stationary solution exists, if are determined by the initial condition from M.
The solution with this initial data exists for all t>0, is unique and converges to the stationary solution.
ξβξα βα
αβ
GG ee KK ,, ~~
constAtf ki
ki
βα
kμ
rn
k
kkG1
0 μξf
kkA
βα
kA
xx G
ξff GGH
ξ
The main calculation
βα
αβ
αβ
fαf
,ii
i GeKdt
df ,~
βα
βα
αβ
αβ
fβfαf
,ii
i GG ee KKdt
df ,, ~~
2
1
ξβξα βα
αβ
GG ee KK ,, ~~
ξff GGH
βα
αβ
αβ
ξfβξfαξαffαβ
f
,
GGGGG eeeKHdt
dH ,,,~,
2
1
0dt
dH 0 yx eexy
The dynamical equilibrium
If is independent on , then we have the system:
The generalization of principle of dynamic equilibrium:
fαβ f
βα
αβ
fα
,ii
i GeKdt
df , ni ,,2,1
αβ
αβ
αβ KK
~
β
βα
β
αβ
ξβξα GG ee KK,,
The time means and the Boltzmann extremals
The Liouville equation
Solutions of the Liouville equation do not converge to the stationary solution. The Liouville equation is reversible equation.
The time means or the Cesaro averages
The Von Neumann stochastic ergodic theorem proves, that the limit, when T tends to infinity, is exist in for any initial data from the same space.
The principle of maximum entropy under the condition of linear conservation laws gives the Boltzmann extremals. We shall prove the coincidence of these values – the time means and the Boltzmann extremals.
xvx dtd nxxx ,,, 21 x
0 xvdivft
f
0xvdiv xgx tftf ,0,
xxxxv nvvv ,,, 21
T
T dttfT
f0
,1
xx
nRL2
Entropy and linear conservation lawsfor the Liouville equation
Let define the entropy by formula
as a strictly convex functional on the positive functions from
Such functionals are conserved for the Liouville equation if
Nevertheless a new form of the H-theorem is appeared in researches of
H. Poincare, V.V. Kozlov and D.V. Treshchev: the entropy of the time average
is not less than the entropy of the initial distribution for the Liouville equation.
Let define linear conservation laws as linear functionals
which are conserved along the Liouville equation’s solutions.
xxx dhhhS ln xx dhhS
0xvdiv
nRL2
hqdhqhI q , xxx
The Boltzmann extremal,the statement of the theorem
Consider the Cauchy problem for the Liouville equation with positive initial data
from . Consider the Boltzmann extremal as the function,
where the maximum of the entropy reaches for fixed linear conservation laws’ constants
determined by the initial data.
The theorem.
Let on the set, where all linear conservation laws are fixed by initial data, the entropy is
defined and reaches conditional maximum in finite point.Then: 1) the Boltzmann extremal exists into this set and unique; 2) the time mean coincides with the Boltzmann extremal.
The theorem is valid and for the Liouville equation with discrete time:
on a linear manifold, if maps this manifold onto itself, preserving measure.
0f nRL2 0fff BB
xφx ,,1 tftf xφ
The case, when
Such functionals are conserved for the Liouville equation:
We can take them as entropy functionals. The solution of the Liouville equation is
Such norm is conserved as well as the entropy functional, so the norm of the linear operator (given by solution of the Liouville equation) is equal to one, and hence the theorem is also valid in this case.
0xvdiv
0 xvdivft
f
x 0xvdiv
xdffS
xgxg
xx t
tftf
,0,
xxx dfSf 2 2FF
fF
0,
x
xvF
t
F
xgx tFtF ,0,
The circular M. Kaс model
Consider the circle and n equally spaced points on it (vertices of a regular inscribed polygon). Note some of their number: m vertices, as the set S. In each of the n points we put the black or white ball. During each time unit, each ball moves one step clockwise with the following condition: the ball going out from a point of the set S changes its color. If the point does not belong to S, the ball leaving it retains its color.
6n 2m
The circular M. Kaс model
Sp if1p
nptp ,,2,1for 1
1p Sp if
Tn tttt ,,, 21 η
The circular M. Kaс model
.,,3,2 ,1
,1
11
1
nptt
tt
ppp
nn
tt Gηη 1
Tn tttt ,,, 21 η
0000
00000
0000
0000
0000
1
2
1
n
n
G
tftf nn ;,,,1;,,, 1322121
.
,1,,3,2,1 ,
1
1
nn
ppp np
The circular M. Kaс model
tt Tff 1
tftf nn ;,,,1;,,, 1322121
1221 ηηηη n
consttftftf n ;;; 221 ηηη
1000
0000
0010
0001
1000
EТ
nt 2dim f
The circular M. Kaс model
TnnnnT
n 112211321 ,,,,,,,, Gηη
TnnnnnnnnT
n 2211121112
321 ,,,,,,,, ηGη
ηGη k
d2
r
rppn 21222
rir
i pp 2122dk 2
dk
knd ,DivisorCommon Greatest the
nk 2 ofdivisor a is
The circular M. Kaс model
kpn
2 if ,2
2
2 if ,2
22
2
22
2
22
2
2
1
2
2
12
p
pppp
k
k
ppppp
k
kk
number. prime a is pn
p
p 22
1
2
2 if ,1
2 if ,2
22
2
2
p
pp
p
:even For m
: oddFor m
k
ppppp
ppp
kk 12
222222
1
22
:even For m : oddFor m
The circular M. Kaс model
3232
2 2 ppppn
2
232
22222222
1
2 22
232
ppp
pppp
3
2232
22
23232
23232 2222222222
pppp
pppppppppp
32
2322
232 2
2222
2
2222
2
22
2
22
2
22
2
2 22
23232
2323222
232
ppppppp
pppppppppppppp
:even For m
: oddFor m
CONCLUSIONS
1. We have proved the theorems which Generalize classical Boltzmann H-theorem quantum case, quantum random walks, classical and quantum chemical kinetics from unique point of vew by general formula for entropy.
2. We have proved a theorem, generalizes Poincare- Kozlov -Treshev (PKT) version of H-theorem on discrete time and for the case when divergence is nonzero.
3. Gibbs method
Gibbs method is clarified, to some extent justified and generalized by the formula
TA = BE Time Average = Boltzmann Extremal
A) form of convergence – TA.B) Gibbs formula exp(-bE) is replaced byTA in nonergodic case.C) Ergodicity: dim (Space of linear conservational laws ) – 1.
New problems
1. To generalize the theorem TA=BE for non linear case (Vlasov Equation).
2. To generalize it for Lioville equations for dynamical systems without invariant mesure (Lorents system with strange attractor)
3. For classical ergodic systems chec up Dim(Linear Space of Conservational Laws)=1.
Thank you for attention