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2003-06

A reduction of the Boltzmann-Peierls Equation

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