Asymptotics of the free path length in the periodic Lorentz gas in the Boltzmann-Grad limit

Abstract

We study the distribution of the free path length in the periodic Lorentz gas in the Boltzmann-Grad limit. For any unimodular Euclidean lattice and any smooth initial velocity distribution, we establish an asymptotic formula for the distribution function of the free path length with a polynomial rate of convergence. The proof combines the dynamical interpretation of the free path length in terms of expanding translates of horospherical orbits on the space of affine lattices, due to Marklof--Strömbergsson, with the effective equidistribution results of Kleinbock--Margulis and Kim. This extends the asymptotic formula of Boca--Zaharescu for the planar case to arbitrary dimension.