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). Htheorem (Boltzmann,1872). Maxwell (18311879) 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.
f i  p i f i 
i  1,2, , n
  ,   Fi  f1 , , f n 
t  mi x 
3
fi t, x h is a distribution function of particles in space point x at a time t,
with mass mi and momentum pi , if f i t , x  is an average number of
3
particles in one quantum state, because the number of states in px is px h
Fi  f1 ,, f n    klij  f k fl 1  fi 1  f j   fi f j 1  f k 1  fl 
k ,l , j
models the collision integral.
  1for fermions,   1 for bosons,   0for the Boltzmann (classical) gas:
Fi  f1 ,, f n    klij  f k fl  fi f j 
k ,l , j
The Carleman model




 df1
2
2


f

f
,
2
1
 dt
 df
 2   f12  f 2 2 ,
 dt
dH f   H f  H f  
 f 2 2  f12  ln  f1   ln  f 2  f 2 2  f12  0
 

dt
f 2 
 f1



H f   S f   f1 ln f1 1  f 2 ln f 2 1
f1  f 2  A  const
d  f1  f 2 
0
dt
Lf , λ   H f     f1  f 2  A

 y  xe x  e y   0
Lf 0 , λ 
Lf 0 , λ 
0
0
f
λ
The Carleman model and its generalizations




 df1
2
2


f

f
2
1 ,
 dt
 df
 2   f12  f 2 2 ,
 dt


 df1
 2
 G f  
 G f   
1

  K 2 exp  2
 ,
  f  K1 exp  2


 dt
 f 2 
 f1  


 df 2   f  K 1 exp  2 G f    K 2 exp  2 G f   ,
1
 f 
 f  
 2
 dt
1
2 





 Gξ  
 Gξ  
1



K exp  2
 K 2 exp  2

 f 2 
 f 1 
2
1


 df1
2
2
2
2






f
1


f

f
1


f
,
2
1
1
2
 dt
 df
 2   f12 1  f 2 2  f 2 2 1  f1 2 .
 dt
The Н-theorem
for generalization of the Carleman model
 Gξ  
 Gξ  
1



K exp  2
 K 2 exp  2

 f 2 
 f 1 
2
1
H f   Gf   Gξ 
H f   Gf   Gξ , f 

 G f  
 G f   
dH f   H f  H f  
 f  K12 exp  2
  K 21 exp  2
  
 



dt
f 2 
 f1
 f 2 
 f1  

  G f  G ξ    G f  G ξ   
 G ξ  
  
  f K12 exp  2
 
  



f1   f 2
f 2  
 f 2 
  f1

  Gf  G ξ   
  Gf  G ξ    

   exp  2
    0
 exp  2




f 2  
f1   
  f 2
  f1

 y  x e x  e y  0


The Markoff process (the random walk)
with two states and its generalizations
 df1
2
1

K
f

K
1 2
2 f1 ,
 dt
 df
 2  K 21 f 1  K 12 f 2 ,
 dt
 df1
2
1



K
f
1


f

K
1 2
1
2 f 1 1  f 2 ,
 dt
 df
 2  K 21 f 1 1  f 2   K12 f 2 1  f 1 ,
 dt

df m
  1  f m 1  f j  K mj h j  K mj hm
dt
j

m  1,, n
hm  f m 1  f m 
Equations of chemical kinetics
df i
   i   i K αf α
β
dt α,β 
df i
   i  K αf α  K βf β 
α 
dt α,β   β
fα

i  1,2, , n
f 1α f 2α  f nα
1
2
Kβα
n
1S1   2 S 2     n S n  1S1   2 S 2     n S n
K K
β
α
K
α
β
 K 
α
β
α
β
β  1 ,  2 , ,  n 
α, β  
 fi


S f    H f    f i  ln  1
i 1
 i

n
S  D E C
α
β
α  1 ,  2 , ,  n 
K
β
α
β
ξ α   K αβξ
β
β
Н-theorem for generalization of
equations of chemical kinetics
df i
~ α, G f 
   i   i  βα f K βα e
dt α,β 
i  1,2,, n
df i
 ~ α, G f  ~ β β, G f  
   i βα f  K βα e
 Kα e

dt α,β 

The generalization of the principle of detailed balance:
~ α, G ξ  ~ β β, G ξ 
Kβα e
 Kα e
Let the system is solved for initial data from M, where G
is defined and continuous.
Let M is strictly convex, and G is strictly convex on M.
 βα f    αβ f 
The statement of the theorem
Let the coefficients of the system are such that there exists at least one solution ξ
in M of generalization of the principle of detailed balance: K~ α eα, Gξ   K~ β eβ, Gξ 
Then:


H f   Gf   Gξ 
α
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
ik fi t   Ak  const ,
where r is the dimension of the linear span of vectors α  β , and vectors μ k
orthogonal to all α  β. Stationary solution is unique, if we fix all the constants
of these conservation laws, and is given by formula

nr

k k 





G
ξ

μ
 

f0

k 1


k

β
G x  x
where the values ​  are determined by A ;
c) such stationary solution exists, if Ak 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.
k
The main calculation
df i
~ α, G f 
   i   i  βα f K βα e
dt α,β 
df i 1
 ~ α, G f  ~ β β, G f  
   i   i  βα f  K βα e
 Kα e

dt 2 α,β 

~ α, G ξ  ~ β β, G ξ 
Kβα e
 Kα e
dH f  1
~ α, G ξ  α, G f   G ξ  β, G f   G ξ 
  β  α, H f  βα f K βα e
e
e


dt
2 α,β 

dH
0
dt
H f   Gf   Gξ 
 y  xe x  e y   0
The dynamical equilibrium
α
If  β f  is independent on f , then we have the system:
df i
α, Gf 
i  1,2,, n
   i   i K βα e
dt α,β 
~
K βα   βα K βα
The generalization of principle of dynamic equilibrium:
 Kβα e
β
α, Gξ    K eβ, Gξ 
β
α
β
The time means and the Boltzmann extremals



The Liouville equation
dx dt  vx
x  x1 , x2 ,, xn 
f
 divfvx   0
divvx  0
t
vx  v1 x, v2 x,, vn x

Solutions of the Liouville equation do not converge to the stationary solution.
The Liouville equation is reversible equation.
T
The time means or the Cesaro averages f x   1 f t , x dt
T



f t , x  f 0, g t x
T

0
The Von Neumann stochastic ergodic theorem proves, that the limit, when T
tends to infinity, is exist in L2 R n 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.
 
Entropy and linear conservation laws
for the Liouville equation

Let define the entropy by formula S h    hx  ln hx dx S h    hx dx

 

as a strictly convex functional on the positive functions from L2 R n

Such functionals are conserved for the Liouville equation if divvx  0

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
I q h    qx hx dx  q, h 
which are conserved along the Liouville equation’s solutions.
The Boltzmann extremal,
the statement of the theorem

Consider the Cauchy problem for the Liouville equation with positive initial data
from L2 R n . Consider the Boltzmann extremal f B  f B f 0
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:
f t  1, x  f t , φx
on a linear manifold, if φx 
maps this manifold onto itself, preserving measure.
f 0
The case, when
divvx  0
f
 divfvx   0
t
 x
divvx  0

F f 
F 
F 
  vx ,
0
t 
x 

S  f      f  dx

We can take them as entropy functionals.
The solution of the Liouville equation is
 F    F 2  f   S 


f t , x  
 x 

 g x 
t

F t , x  F 0, g t x
Such functionals are conserved for the Liouville equation:

f 0, g t x 
f 2 x  xdx
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.
The circular M. Kaс model
n6
m2
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.
The circular M. Kaс model
 p t   1 for p  1,2,, n
ηt   1t ,2 t ,,n t 
T
 p  1 if p  S
 p  1 if p  S
The circular M. Kaс model
1 t  1   nn t ,
 p t  1   p 1 p 1 t , p  2,3,, n.
ηt   1t ,2 t ,,n t 
T
ηt 1  Gηt 
 0 0 0  0 n 



0
0

0
0
 1

0 
0  0
0
2

G 
      
0 0 0  0

0


0 0 0  
0 
n 1

 p   p p 1 , p  1,2,3,, n  1,
 n   n1.
f 1,2 ,,n ; t  1  f 12 ,  23 ,,  n1; t 
The circular M. Kaс model
f 1,2 ,,n ; t  1  f 12 ,  23 ,,  n1; t 
f t 1  Tf t 
dimf t   2n
η1  η2    η2n  η1
Т  E 
  



 

 1
 0


 0

 0

f η1; t  f η2 ; t   f η2n ; t  const
0
0

0

0

0
1
 
0




0
0
 
0
0

1
1 

0 
0 


0 
  
The circular M. Kaс model
η  Gk η
1,2 ,3 ,,n T  η  Gη   nn , 11,  22 ,,  n1n1 T
1,2 ,3 ,,n T  η  G2η   n n1n1, 1 nn ,  211,,  n1 n2n2 T
d  theGreatest CommonDivisorn, k 
k is a divisor of 2n

2n  21  p2 2   pr
r
k d

k  2d  2 1  p2 2  pr r
i
i
2d
The circular M. Kaс model
n  p is a prime number.
For even m :
2 2p  2

, if p  2
For odd m :
2
2p
1, if p  2
2 2 2

1
p
p
n  pk
For even m :
For odd m :
2 2p  2 2p  2p
2p  2p


 
2
1
p
p
pk
2
k
2 2 2 2 2
2 2





2
2p
2 p2
2 pk
p
k
22
, if p  2
k 1
2
p2
p
pk
k 1
p k 1
, if p  2
The circular M. Kaс model
n  p2  p3
2
2  p2  p3 
For even m :
2 2 2 2 2 2



1
p2
p3
p3
p2

p2 2
2
p 2 p3


 2 p2  2  2

2
p2

 

 2 2  2 2 2 2

p2 p3
p 2 2 p3
 2 p 2 p3  2 p 2  2 p 2
2
p2 p3
2 2 2 2

2 p2 p3
p 2 2 p3
2
2
2
2 p2 p3
p2
p3
2
For odd m :
p2 2
2 2 2 2 2 2 2



2
2
2 p2
2 p3
2 p2
p2
p3
p2

2
p 2 p3
p2
p3
p2 p3
p2 2
 2 p2
CONCLUSIONS
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.
1.
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