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