Diffusion driven by collisions with the boundary

Article type: Research Article

Authors: Degond, P. | Mancini, S.

Affiliations: Mathématiques pour l'Industrie et la Physique, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France E‐mail: [email protected]‐tlse.fr | Dipartimento U. Dini, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy E‐mail: [email protected]

Abstract: We present a mathematically rigorous derivation of a diffusion model previously introduced by the first author to model the diffusion of charged‐particles moving in the gap between two plane parallel plates. The particles are subject to crossed electric and magnetic fields and to collisions against the surface of the solid plates. The surface collisions are supposed to be elastic. Under appropriate scaling assumptions, the particle distribution function converges to a function of the energy and of the longitudinal position coordinates only, which evolves in time according to a diffusion equation. A rigorous convergence proof is given. The proof relies on precise estimates on the trace of the distribution function at the boundary.

Keywords: diffusion approximation, transport equation, accomodation boundary condition, transport–diffusion equation, spherical harmonics expansion model, moment method, Hilbert expansion, Onsager relation

Journal: Asymptotic Analysis, vol. 27, no. 1, pp. 47-73, 2001