On the distribution of free path lengths for the periodic Lorentz gas II

Consider the domain Z(epsilon) = {x is an element of R-n ; dist(x, epsilonZ (n)) > epsilon gamma} and let the free path length be defined as tau (epsil on)(x, v) = inf{t > 0 ; x - tv is an element of partial derivativeZ(epsilon )}. In the Boltzmann-Grad scaling corresponding to gamma = n/n-1, it is sho wn that the limiting distribution phi (epsilon) of tau (epsilon) is bounded from below by an expression of the form C/t, for some C > 0. A numerical s tudy seems to indicate that asymptotically for large t, phi (epsilon) simil ar to C/t. This is an extension of a previous work [J. Bourgain et al., Com m. Math. Phys. 190 (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe the Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usua l case of a Poisson distribution of scatterers treated in [G. Gallavotti (1 972)].