six lectures about (advanced) statistical...
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LECTURES ABOUT
(ADVANCED) STATISTICAL
PHYSICS
T.S.Biró, MTA Wigner Research Centre for Physics, Budapest
Lectures given at: University of Johannesburg, South-Africa,
November 26 – November 29, 2012.
1. Ancient Thermodynamics (… - 1870)
2. The Rise of Statistical Physics (1890 – 1920)
3. Modern (postwar) Problems (1940 – 1980)
4. Corrections (1950 – 2005)
5. Generalizations (1960 – 2010)
6. High Energy Physics (1950 – 2010)
2
LECTURE TWO ABOUT
(ADVANCED) STATISTICAL
PHYSICS
T.S.Biró, MTA Wigner Research Centre for Physics, Budapest
Lectures given at: University of Johannesburg, South-Africa,
November 27, 2012.
Kinetic theory
• Sum of random forces: noise (Gaussian)
• Brownian motion
• Langevin and Fokker-Planck equations
• Fluctuation-dissipation theorem
• Boltzmann equation
• Entropy equilibrium theory
TSB, CG, PRL 79, 3138, 1997
TSB, PG, hep-ph/0503204
General Langevin problem
p = F ( p, z ) .
Wang + Uehlenbeck:
use R(p) test function!
Many p(t) evolutions from p(0): f(p,t) distribution
),()),((),()( tpfzpFdtpRdpdttpfpRdp
average over noise < F > = - G(p),
< F F > - < F > < F > = 2 D(p) / dt
General Langevin problem
Expansion til o(dt) gives:
),()()()()()( tpfpDpRpGpRdp
t
fpRdp
Fokker-Planck equation after partial integration:
fpDp
fpGpt
f)()(
2
2
Particular Langevin problem
p = z - G(E) ∂E
∂p
. < z(t) > = 0
< z(t)z(t') > = 2 D(E) (t-t')
In the Fokker – Planck equation: D (p) = D(E)
G (p) -G(E) ∂E
∂p Stationary distribution:
f(p) = exp - G(E) ∫ D(E)
dE
D(E)
A ( )
=
TSB, GGy ,AJ, GP, JPG31, 759, 2005
)(
expET
dEA
Inverse logarithmic slope temperature
T(E)
1 = ln f (E)
d
dE
T (E) = D(E)
G(E) + D'(E)
T = D(0) / G(0) Gibbs
T = D(E) / G(E) Einstein
General inverse slope
Stationary distribution:
f(p) = A exp - ∫ T(E)
dE
1) Gibbs: T(E) = T exp(-E/T)
2) Tsallis: T(E) = T/q + (1-1/q) E
( 1 + (q-1) E / T) -q /(q-1)
( )
T( T ) = T : a fixed point of the sliding slope
Fluctuation Dissipation theorem
D (E) = 1
f(E)
with f(E) stationary distribution
∫ E
∞
G (x) f(x) dx ij ij
D (E) = T(E) ij
G (E) + ij
D' (E) ij ( )
(Hamiltonian eom does not change energy E!)
p = -G E + z i i j ij
.
Fluctuation Dissipation theorem
particular cases ( for constant G ):
D = T ij
G ij
D (E) = T + (q-1) E ij
G ij ( )
Gibbs:
Tsallis:
Field theory calculation
• polynomial interaction, one field integrated out
• Imaginary part of self-energy noise
• Effective Langevin eq. for soft field
• Fluctuation-dissipation: 𝑫𝒊𝒋 = 𝑻𝟎 𝑮𝒊𝒋 (constant Einstein temperature)
• G is linear in the energy: 𝑮𝒊𝒋 = 𝜸𝒊𝒋(𝟏 + 𝑬/𝑻𝟎𝑪𝟎)
• f(E) Gaussian
𝟏
𝕿(𝑬)=
𝟏
𝑻𝟎+
𝟏
𝑪𝟎𝑻𝟎 + 𝑬
slope
c
T
𝕿 𝟎 = 𝑪𝟎𝑻𝟎
𝑪𝟎 + 𝟏
E E
𝕿 ∞ = 𝑻𝟎
Einstein
Gibbs
Walton – Rafelski (PRL 2000)?
𝟏
𝕿(𝑬)=
𝟏
𝑻𝟎+
𝟏
𝑪𝟎𝑻𝟎 + 𝑬
𝔗 𝐸 ≈ 𝑇 0 + 𝐸
𝐶0 + 1 2+ ⋯
Additive and multiplicative noise
1. Langevin
p = - p = G F
2C 2B 2D
2. Fokker Planck
∂f ∂t
∂ ∂p
∂ ∂p
= ( K f ) - ( K f ) 1
2
2 2
K = F – Gp
K = D – 2Bp + Cp 2 2
1
c c c
Equivalent descriptions: TSB, AJ, PRL 94, 132302, 2005
Exact stationary distribution:
f = f (D/K ) exp(- atan( ) ) 0
v 2 D – Bp
p
with v = 1 + G/2C
= GB/C – F
= DC – B 2 2
For F = 0 characteristic scale: p = D/C. c 2
power
exponent
(small or large) parameter
Exact stationary distribution for F = 0, B = 0:
f = f ( 1 + ) 0
-(1+G/2C) 2
D
C p
With E = p / 2m this is a Tsallis distribution!
f = f ( 1 + (q-1) ) 0
E
2
T
q
1 – q
Tsallis index: q = 1 + 2C / G Temperature: T = D / mG
Limits of the Tsallis distribution:
p p : Gauss
p p : Power-law
c
f ~ exp( - Gp /2D )
f ~ ( p / p )
2
c
-2v
c
E E :
E E :
c f ~ exp( - E / T )
f ~ (E / E ) -v
c c
Relation between slope, inflection and power !!
v = 1 + E / T c
Energy distribution limits:
Stationary distributions
For F=0, B=0 the Tsallis distribution is the exact stationary solution
Gamma: p = 0.1 GeV
F ≠ 0
Gauss: p = ∞
Zero: p = 10 GeV
B = D/C
Power: p = 1 GeV
F ≠ 0
c
c
c
c
2
MODERN PROBLEMS
o Information
o Chaos
o Phase Transition
o Scaling
o Anomalous Fluctuations
20
Neumann
Shannon
Rényi
Wilson
Lévy
Feynman
Lyapunov
Kolmogorov
Tsallis
Statistics, entropy, temperature
Fermi distribution (Bernoulli, Poisson)
Bose distribution (negative binomial)
Superstatistics: distribution of distributions
21
Fermi distribution: N, K-N
N1NN1N
BmaxN)21N(Wlnk
N
K
)!NK(!N
!KW
Fermi distribution: N, K-N
N
1NKx
1N
NK
xlnln
xlnln
.1k),(xln :Notation
N1NK
1NN
1NNK
N1N
B
Fermi distribution: N, K-N
x1
1
K
1
x1
1f
x1
x
K
1
x1
1
f
K1f1x
K1f
f1
K,K/Nf :fix
Fermi distribution: N, K-N
1e
1w
x1
1)x(wf
)(Fermi
Fermi
Fermi distribution in a subsystem
N
K
nN
kK
n
k
Pk,n
Fermi distribution in small subsystems
)nk(n
k
k,n
n
)NK()!NK(N!N
K!K
N
K
n
k
P
N!N)!nN(,Nn,Kk
Fermi distribution in small subsystems = Bernoulli distribution
nkn
k,n
nk
k,n
)f1(fn
kP
NK
N
K
NK
n
kP
A story of false coins
Bose distribution in a sunsystem
N
1NK
nN
nNkK
n
nk
Pk,n
k levels and n excitations mixed arbitrarily…
Bose distribution in small subsystems
1e
1wf
f)1k(n
)f1(fn
nkP
)(Bose
n1kn
k,n
Negative binomial distribution (NBD)
n1kn
k.n
n
)f1()f(n
1kP
n
1k)1(
n
nk
Fermi – Bose transformation: statistical supersymmetry
)1k(kinvariant
)f(B)f(F
)f(F)f(B
1k,nk,n
1k,nk,n
Rare events: Poisson distribution
an
an
e!n
aP
2
a
n
n
Rare event Bernoulli: Poisson
nx
kn
n
k
n
n
ke!n
1)x(CP
f1
fk
!n
1)f1(P
!n
k
n
kkn
Rare event NBD: Poisson
nx
kn
n
k
n
n
ken
xCP
f
fk
nfP
n
k
n
nkkn
!
1)(
1!
1)1(
!
1
CORRECTIONS
o Finite Size Effects
o Near-Equilibrium Fluctuations
o Scaling Fluctuations
o Superstatistics
36
NBD = Euler ○ Poisson
0
x)f1(nk
n
n1kn
k,n
1N
0
axN
dxex!n!k
f
)f1(fn
nkP
a
!Ndxex
NBD = Euler ○ Poisson
0
x
k
xf
n
k,ndxe
!k
xe
!n
)fx(P
Poisson in k, Euler-Gamma in x
S u p e r s t a t i s t i c s
max: 1 – 1/c, mean: 1, spread: 1 / √ c
Euler - Gamma distribution
Homework problems
1. Regard the following distributions:
– Bernoulli ( n, k; f )
– NBD ( n, k; f )
– Poisson ( n, k; f )
Questions:
– Check the norm
– n expectation value, squared variance
– characteristic function (expectation value of exp(bn) )
Homework problems
1. How is the supertransformation for finite
subsystems inside finite systems ?
B( n; k | N; K ) F( n; k | N; K )
What is the expectation value of f(a+x), if x
is distributed as
a) Gauss
b) Euler-Gamma ?
