A REDUCTION OF THE BOLTZMANN-PEIERLS EQUATION

Abstract

This paper deals with the solutions of initial value problems of the Boltzmann-Peierls equation (BPE). This integro-differential equation describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE. By using special coordinates in the one-dimensional case, we can perform a further reduction of the kinetic equation. By assuming one-dimensionality on the initial phase density one can show that this property is preserved for all later times. We derive kinetic schemes for the kinetic equation as well as for the derived moment systems. Several numerical test cases are presented to validate the theory.