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

TL;DR

This paper derives the linear Landau and Boltzmann equations with magnetic effects from a Lorentz model in different regimes, providing explicit error estimates and comparing with hard disk systems to analyze memory effects.

Contribution

It establishes the derivation of kinetic equations with magnetic fields from a Lorentz model, including explicit error bounds and comparison with hard disk systems.

Findings

Linear Landau equation derived in weak coupling limit with magnetic field.

Linear Boltzmann equation derived in low density regime with inverse power law potentials.

Memory effects are negligible in the Lorentz model but significant in hard disk systems.

Abstract

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 a magnetic transport term. 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 a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in \cite{BMHH}, which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.