A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions

We rigorously derive a linear kinetic equation of Fokker-Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obsta cle of the Lorentz gas generates a potential epsilon V-alpha(\x \/epsilon), where V is a smooth radially symmetric function with compact support, and alpha > 0. The density of obstacles diverges as epsilon (-s), where delta > 0. We prove that when 0 < alpha < 1 /8 and delta = 2 alpha + 1, the probab ility density of a test particle converges as epsilon --> 0 to a solution o f our kinetic equation.