Two-dimensional Lorentz process for magnetotransport: Boltzmann-Grad limit

Alessia Nota (1) , Chiara Saffirio (2) , Sergio Simonella (3, 4)

1 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
2 Department of Mathematics [Basel]
3 CNRS - Centre National de la Recherche Scientifique
4 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We study a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the effect of a magnetic field perpendicular to the plane. We prove that, in the low-density (Boltzmann-Grad) limit, the particle distribution evolves according to a generalized linear Boltzmann equation, previously derived and solved by Bobylev et al. [BMHH, BMHH1, BHPH]. In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of [Ga69] can be however adapted to prove convergence of the process with memory.

Mots clés: low-density limit; generalized Boltzmann equation; magnetic field; Lorentz gas; memory terms; non- Markovian process

Domaines: Mathématiques [math]

Alessia Nota, Chiara Saffirio, Sergio Simonella. Two-dimensional Lorentz process for magnetotransport: Boltzmann-Grad limit. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2022, 58 (2), pp.1228-1243. ⟨hal-02965043v2⟩