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

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael
- Kunik, Matthias

*2010 Mathematics Subject Classification*

- 82C40 82C70

*Keywords*

- Kinetic theory of phonons, Maximum Entropy Principle

*DOI*

*Abstract*

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 with respect to the following questionings: In thermodynamics, a given kinetic equation is usually replaced by its truncated moment systems which in turn is supplemented by a closure principle so that there results a system of PDE's for some moments as thermodynamic variables. A very popular closure principle is the Maximum Entropy Principle yielding a symmetric hyperbolic system. In recent times this strategy has lead to serious studies on two problems that might arise. 1. Do solutions of the Maximum Entropy Principle exist? 2. Is the physics which is embodied in 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/Kunik. The current study yields a positive existence result. We prove for the Boltzmann-Peierls-Equation hat the Maximum Entropy Principle is well posed and that it can be used to establish a closed hyperbolic moment system of PDE's. Regarding the second question on the equivalence of moments that are calculated by solutions of the Boltzmann-Peierls-Equation and moments that result from the Maximum Entropy system we develop a numerical method that allows a comparison of both solutions. In particular, we introduce a numerical kinetic scheme that consists of free flight periods and two classes of update rules. The first class of rules are the same for the kinetic equation as well as for the Maximum Entropy system, while the second class of update rules contain additional rules for the Maximum Entropy system. It is illustrated that if sufficient many moments are taken into account, both solutions converge to each other. However, it is additionally illustrated, that the numerical effort to solve the kinetic equation is less than the effort to solve the Maximum Entropy system.

*Appeared in*

- Contin. Mech. Thermodyn. 16 (2004), pp. 453--469

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