On the derivation of a linear Boltzmann equation for a periodic lattice gas

We consider the problem of deriving the linear Boltzmann equation from the Lorentz process with hard spheres obstacles. In a suitable limit (the Boltzmann-Grad limit), it has been proved that the linear Boltzmann equation can be obtained when the position of obstacles are Poisson distributed, while the validation fails, also for the "correct" ratio between obstacle size and lattice parameter, when they are distributed on a purely periodic lattice, because of the existence of very long free trajectories. Here we validate the linear Boltzmann equation, in the limit when the scatterer's radius epsilon vanishes, for a family of Lorentz processes such that the obstacles have a random distribution on a lattice and the probability for an obstacle to be on a given lattice site p = epsilon(delta/(2 -delta)) and the lattice parameter l = epsilon(1/(2 - delta)), 0 < delta less than or equal to 1, are related to the radius epsilon according to the Boltzmann-Grad scaling.