Symmetry of the Linearized Boltzmann Equation and Its Application

A symmetric relation of macroscopic quantities between two different steady problems of the linearized Boltzmann equation is derived. A few applications to half-space problems are presented first. Then, for the gas in bounded or unbounded domains such that solid bodies or condensed phases are confined in a finite region, general representations of the mass, momentum, and heat fluxes through the boundary (possibly at a point on or on a part of it) are derived from the symmetric relation linked to the separability of boundary data. This result implies a reduction of the original problem to a single elemental problem in the same domain, as far as the fluxes are concerned. Many applications are also presented.