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

Consider the domain Z_∈ = {cursive Greek chi ∈ ℝⁿ ; dist(cursive Greek chi, ∈ℤⁿ) > ∈^γ} and let the free path length be defined as τ_∈(cursive Greek chi, v) = inf{t > 0 ; cursive Greek chi - tv ∈ ∂Z_∈}. In the Boltzmann-Grad scaling corresponding to γ = n/n-1, it is shown that the limiting distribution φ_∈ of τ_∈ is bounded from below by an expression of the form C/t, for some C > 0. A numerical study seems to indicate that asymptotically for large t, φ_∈ ∼ C/t. This is an extension of a previous work [J. Bourgain et al., Comm. 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 usual case of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].