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

describes the evolution of a density of particles in a medium in which the particles don’t interact among themselves. The motion of each particle is described by a jump process: The speed of a particle is constant (equal to one), and also the direction is constant in exponentially distributed time intervals. At the end of such an interval, the direction jumps according to a law that corresponds to the specular reflection on a circular obstacle (with a uniformly distributed impact parameter). This Boltzmann equation can be rigorously derived as the “Boltzmann-Grad” limit of a system with obstacles of finite size. This was done by Gallavotti [Ga1, Ga2] (but see also Spohn [Sp]) by considering obstacles of diameter whose centers are distributed in the plane according to a Poisson law with density −1. A formal calculation yields a mean free path of order one, uniformly in , and Gallavotti showed that this is rigorously true, and that the limiting evolution equation is really the Boltzmann equation (1). Quite contrary to this, Bourgain et al. [BGW, GW] showed that the corresponding scaling for a periodic distribution of scatterers cannot give rise to a Boltzmann equation, the reason being that the distribution of free path lengths is not exponential in that case. An asymptotic formula is given in [CG]. At a formal level, however, it can still work as was shown by Golse [G]. As a way of deriving a linear Boltzmann equation starting from a periodic distribution of scatterers, Caglioti et al [CPR] considered scatterers of diameter with centers on a rectangular lattice with parameter : in each lattice point, independently of the other lattice points, the probability of finding a scatterer is . In the limit as tends to zero, this distribution approaches a Poisson distribution, but one cannot immediately infer from that, that the dynamics of scattered particles approach a Boltzmann process.