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Diffusive Limit of a Two Dimensional Kinetic System of Partially Quantized Particles

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Abstract

A quantum-classical coupled system which models the diffusive transport of electrons partially confined in semiconductors nanostructures was presented in Ben Abdallah and Méhats (Proc. Edinb. Math. Soc. 49:513–549, 2006). In this model, electrons are assumed to behave like wave in the confinement direction and to have a classical behaviour in a diffusive regime in the transport direction parallel to the electron gas. It was formally derived from a kinetic system for partially quantized particles thanks to a diffusive limit when the mean free path becomes small with respect to the macroscopic length scale. This paper is devoted to the rigorous study of this limit for a transport in one dimension. In the transport direction, the motion of particles is described by a 1D Boltzmann equation. A Boltzmann-Schrödinger-Poisson system is then considered. Existence of renormalized solutions relying on the study of a quasistatic Schrödinger-Poisson system and on an entropy estimate is established. Its diffusive limit is then considered.

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  1. Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, BC 187, 4 place Jussieu, 75252, Paris Cedex 5, France

    Nicolas Vauchelet

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  1. Nicolas Vauchelet
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Vauchelet, N. Diffusive Limit of a Two Dimensional Kinetic System of Partially Quantized Particles. J Stat Phys 139, 882–914 (2010). https://doi.org/10.1007/s10955-010-9970-3

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