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.

Automated summary

In this study, we investigated a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the influence of a perpendicular magnetic field. We proved that, in the low-density limit, the particle distribution follows a generalized linear Boltzmann equation, which includes non-Markovian terms. By adapting the ideas from (Phys. Rev. 185 (1969) 308-322), we were able to demonstrate the convergence of the process with memory.