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

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. (Phys. Rev. Lett. 75 (1995) 2, J. Stat. Phys. 87 (1997) 1205-1228, J. Stat. Phys. 102 (2001) 1133-1150). In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of (Phys. Rev. 185 (1969) 308-322) can be however adapted to prove convergence of the process with memory.