The Boltzmann equation as limit of a Schrödinger equation with random potential

Article type: Research Article

Authors: Brassart, Matthieu

Affiliations: Departement de Mathematiques, UMR 6623, Université de Franche Comte, 16 route de Gray, 25030 Besancon cedex, France E‐mail: [email protected]‐fcomte.fr

Abstract: We study the transport of particles in time‐dependent random media in the so‐called weak coupling limit. We show how a limiting irreversible dynamic of Boltzmann's type can be derived from a reversible dynamic of Schrödinger's type. The homogenization is performed at the level of Wigner transforms for a mixed states formulation. We obtain a simple modelization of the influence of microscopic chaos on physical observables by means of a space–time Fourier transform of a covariance associated with the random variations.

Keywords: semi‐classical limit, Wigner transform, Schrödinger equation, Boltzmann equation, random media, weak coupling limit

Journal: Asymptotic Analysis, vol. 37, no. 3-4, pp. 189-214, 2004