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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).
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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:
The Markoff process(the random walk)with two states and its generalizations
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, where is defined and continuous. LetMis strictly convex, and G is strictly convex on M.
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 fromM. The solution with this initial data exists for all t>0, is unique and converges to the stationary solution.
The dynamical equilibrium Ifis independent on , then we have the system: The generalization of principle of dynamic equilibrium:
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 orthe 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.
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.
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 entropyis defined and reaches conditional maximum in finite point. Then: 1) the Boltzmann extremalexists 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.
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.
The circularM. 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.
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.
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 • 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.
