Entropy in the sense of Boltzmann and Poincare, Boltzmann extremals, and the Hamilton–Jacobi method in non-Hamiltonian context

In this paper, we prove the $H$-theorem for generalized chemical kinetics equations. We consider 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 averages coincide with Boltzmann extremals for all such equations and for the Liouville equation as well. This gives us an approach for choosing the action–angle variables in the Hamilton–Jacobi method in a non-Hamiltonian context. We propose a simple derivation of the Hamilton–Jacobi equation from the Liouville equations in the finite-dimensional case.