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Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

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Abstract

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

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Acknowledgments

The author thanks Leif Arkeryd for proposing this study and Alexander Bobylev for encouraging it.

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  1. Department of Mathematics and Computer Science, Karlstad University, 651 88, Karlstad, Sweden

    Niclas Bernhoff

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  1. Niclas Bernhoff
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Correspondence to Niclas Bernhoff.

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Bernhoff, N. Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation. J Stat Phys 159, 358–379 (2015). https://doi.org/10.1007/s10955-015-1190-4

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  • DOI: https://doi.org/10.1007/s10955-015-1190-4

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