?-THEOREM and ENTROPY over BOLTZMANN and POINCARE

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?-THEOREM and ENTROPY over BOLTZMANN and
POINCARE

• Vedenyapin V.V.,
• Adzhiev S.Z.

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?-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.
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The discrete velocity models of the Boltzmann
equation and of the quantum kinetic equations

• We consider the ?-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

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The Carleman model
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The Carleman model and its generalizations
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The ?-theoremfor generalization of the Carleman
model
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The Markoff process (the random walk)with two
states and its generalizations
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Equations of chemical kinetics
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?-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,
  where
• is defined and continuous.
• Let M is strictly convex, and G is strictly
  convex on M.

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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
  tgt0, is unique and converges to the stationary
  solution.

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The main calculation
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The dynamical equilibrium

• If is independent on , then we
  have the system
• The generalization of principle of dynamic
  equilibrium

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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.

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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
  equations solutions.

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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.

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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.

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The circular M. Ka? 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.

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The circular M. Ka? model
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The circular M. Ka? model
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The circular M. Ka? model
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The circular M. Ka? model
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The circular M. Ka? model
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The circular M. Ka? model
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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.

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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.
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New problems

• To generalize the theorem TABE 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.

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Thank you for attention