Statistical Properties of Two-Dimensional Periodic Lorentz Gas with Infinite Horizon

We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the time t goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t log t) 1/2 and not t t/2 as in the classical case. We find the covariance matrix of the limit distribution.