Kinetic transport in the two-dimensional periodic Lorentz gas

Abstract

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann–Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation—in contrast to the Lorentz gas with a disordered scatterer configuration. This paper focuses on the two-dimensional set-up and reports an explicit, elementary formula for the collision kernel of the transport equation.