by Kunik, M., Shamsul, Q., Warnecke, G..

**Series:** 2003-06, Preprints

- MSC:
- 65M99 None of the above, but in this section
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics

**Abstract:**

This paper is concerned 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-dimensinal

kinetic equation which has a much simpler structure than the original BPE. Using

special coordinates in the one-dimensional case, we can perform a further reduction

of the kinetic equation. Making a one-dimensionality assumption 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 shown in order to validate the theory

**Keywords:**

Boltzmann-Peierls equation, Bose-gas, phonos, kinetic schemes