On the distribution of free path lengths for the periodic Lorentz gas II
François Golse, Bernt Wennberg · ESAIM Mathematical Modelling and Numerical Analysis · 2000
Consider the domain and let the free path length be defined as In the Boltzmann-Grad scaling corresponding to , it is shown that the limiting distribution of 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, . 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)].