Magnetotransport in the two-dimensional Lorentz model: non-markovian Grad limit of the BBGKY hierarchy

Abstract

We consider the classical two-dimensional Lorentz model of non-interacting charged point particles acted upon by a perpendicular uniform magnetic field. Particles are elastically scattered by hard disks randomly distributed in the motion plane. Trajectories of gas particles between collisions are arcs of cyclotron circles. We consider the Grad limit of the corresponding BBGKY hierarchy. The Lorentz model in magnetic field provides an example where the Grad limit of the solution for the one-particle density cannot be found by formally applying the limit to evolution equations. The resulting non-markovian kinetic equation is shown to coincide with the generalization of the Boltzmann equation originally proposed upon intuitive arguments.