Power-Law Distributions for the Free Path Length in Lorentz Gases

Abstract

It is well known that, in the Boltzmann–Grad limit, the distribution of the free path length in the Lorentz gas with disordered scatterer configuration has an exponential density. If, on the other hand, the scatterers are located at the vertices of a Euclidean lattice, the density has a power-law tail proportional to $$\xi ^{-3}$$ . In the present paper we construct scatterer configurations whose free path lengths have a distribution with tail $$\xi ^{-N-2}$$ for any positive integer $$N$$ . We also discuss the properties of the random flight process that describes the Lorentz gas in the Boltzmann–Grad limit. The convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces, which in turn is a consequence of Ratner’s measure classification theorem.