On the generalized problem of the Boltzmann equation and the moment method in kinetic theory of gases

Abstract

In this paper we formulate the generalized problem of the Boltzmann equation based on the kinetic entropy balance equation in conjunction with the maximum entropy principle. First we prove that the solution of this generalized problem is unique. We then prove that the entropy balance equation obtained by Eu in extended irreversible thermodynamics is valid, if, and only if, the one-particle distribution function fa is the solution of this generalized problem. As a by-product of this result, we also obtain a statistical expression of the thermodynamic entropy balance equation that shares the same formula as the kinetic entropy balance equation.