The H-theorem for the physico-chemical kinetic equations with explicit time discretization

There is demonstrated in the present paper, that the H-theorem in the case of explicit time discretization of the physico-chemical kinetic equations, generally speaking, is not valid. We prove the H-theorem, when the system of the physico-chemical kinetic equations with explicit time discretization has the form of non-linear analogue of the Markov process with doubly stochastic matrix, and for more general cases. In these cases the proof is reduced to the proof of the H-theorem for Markov chains. The simplest discrete velocity models of the Boltzmann equation with explicit time discretization -the Carleman and Broadwell models are discussed and the H-theorem for them in the case of discrete time is proved.