Symmetry of the Linearized Boltzmann Equation II Entropy Production and Onsager-Casimir Relation

Abstract

This is the second part of the study by the author on the symmetry of the linearized Boltzmann equation. The issue of the present part is the entropy production and the Onsager–Casimir reciprocity relation in the steady non-equilibrium systems. After the discussions on the definition of the entropy, entropy flow, and entropy production in the non-equilibrium gas systems, the expression of the entropy production in the steady state is presented. Then, for the systems weakly perturbed from a uniform equilibrium state, the entropy production is shown to be expressed in terms of the solution of the linearized Boltzmann equation. The thermodynamic forces and fluxes and the kinetic coefficients are defined solely from the expression of the entropy production. The conventional-type Onsager–Casimir relation is shown to hold for the entire range of the Knudsen number in bounded- and unbounded-domain systems, provided that the state of the gas in a far field is a local Maxwellian satisfying the Boltzmann equation for the latter. As to the other unbounded-domain systems, a nonconventional reciprocity is shown to hold.