Derivation of a Linear Boltzmann Equation for a Lattice Gas

#### E. Caglioti, M. Pulvirenti, V. Ricci

2000, v.6, Issue 3, 265-285

ABSTRACT

We consider a Lorentz gas in the plane where the scatterers have random positions on a square lattice. The scatterers are identical disks of diameter $\varepsilon$, which is also the size of the side of a cell and the probability, for a given cell, to be occupied by a scatterer. A point particle moves freely between the scatterers, interacting with them through elastic collisions. We show that, when $\varepsilon \to 0$, the probability density of such a light particle converges to the solution of the linear Boltzmann equation with the hard-sphere cross section.

Keywords: Lorentz gas,Boltzmann equation,jump process