Derivation of the linear Laundau equation and linear Boltzmann equation from the Lorentz model with magnetic field

Matteo Marcozzi, Alessia Nota

Published 2015-07-09Version 1

We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with the magnetic field. Moreover we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with the magnetic field. We provide an explicit control of the error in the kinetic limit. The explicit estimates of the sets of bad configurations preventing the Markovianity are the technical core. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH1}, \cite{BMHH2} which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.