Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

Dreyer, W.; Herrmann, M.; Kunik, M.
July 2004
Continuum Mechanics & Thermodynamics;Jul2004, Vol. 16 Issue 5, p453
Academic Journal
The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons. In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDE�s results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik. This study has two main objectives: 1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDE�s. However, the intent of the paper is by no means a general justification of the maximum entropy principle. 2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.


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