Symmetry of the Unsteady Linearized Boltzmann Equation in a Fixed Bounded Domain

Abstract

A symmetric relation between time-dependent problems described by the linearized Boltzmann equation is obtained for a gas in a fixed bounded domain. General representations of the total mass, momentum, and energy in the domain, as well as their fluxes through the boundary, in terms of an appropriate Green function are derived from that relation. Several application examples are presented. Similarities to the fluctuation–dissipation theorem in the linear response theory and its generalization to gas systems of arbitrary Knudsen numbers are also discussed. The present paper is an extension of the previous work of the author (Takata in J. Stat. Phys. 136: 751–784, 2009) to time-dependent problems.