Irreducible tensor description. III. Thermodynamics of a low-temperature phonon gas

Abstract

Let one assume that the interacting phonon gas, whose behavior is governed by the Boltzmann–Peierls equation, inhabits an insulating crystal at sufficiently low temperature. Then, within the framework of a single acoustic phonon branch and of an isotropic long-wavelength approximation to the dispersion relation, the simplest acceptable version of extended irreversible thermodynamics, based upon the nine-moment system of differential equations for the slow and fast gas-state variables, is carefully investigated. It is clearly demonstrated that, in virtue of the structure simplifications just mentioned, the conceptually different (macroscopic, kinetic, and variational) procedures, which are discussed in this paper, appear complementary to each other. Finally, with the help of a suitable contraction of the nine-moment system of field equations, for Callaway’s relaxation model a slightly generalized nonlinear variant of ordinary low-temperature phonon hydrodynamics is explicitly derived.