Boltzmann and Poincaré Entropy, Boltzmann Extremals, and Hamilton–Jacobi Method for Non-Hamiltonian Situation

Abstract

In this paper, we prove the H-theorem for generalized chemical kinetics equations. We consider the following important physical examples of such a generalization: discrete models of quantum kinetic equations (Uehling–Uhlenbeck equations) and a quantum Markov process (quantum random walk). We prove that time means coincide with Boltzmann extremals for all such equations and for the Liouville equation as well. This gives us an approach to select the action–angle variables in the Hamilton–Jacobi method for the non-Hamiltonian situation. We propose a simple derivation of the Hamilton–Jacobi equation from the Liouville equations in the finite-dimensional case.