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It is well known that, in the Boltzmann-Grad limit, the distribution of thefree path length in the Lorentz gas with disordered scatterer configuration hasan exponential density. If, on the other hand, the scatterers are located atthe vertices of a Euclidean lattice, the density has a power-law tailproportional to xi^{-3}. In the present paper we construct scattererconfigurations 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 flightprocess that describes the Lorentz gas in the Boltzmann-Grad limit. Theconvergence of the distribution of the free path length follows fromequidistribution of large spheres in products of certain homogeneous spaces,which in turn is a consequence of Ratner's measure classification theorem.

Power-law distributions for the free path length in Lorentz gases