Stochastic One-Dimensional Lorentz Gas on a Lattice

We study a one-dimensional stochastic Lorentz gas where a light particle moves in a fixed array of nonidentical random scatterers arranged in a lattice. Each scatterer is characterized by a random transmission/reflection coefficient. We consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables. A symbolic program is presented which generates the exact velocity autocorrelation function (VACF) in terms of the moments of the transmission coefficients. The VACF is found for different types of disorder for times up to 20 collision times. We then consider a specific type of disorder: a two-state Lorentz gas in which two types of scatterers are arranged randomly in a lattice. Then a lattice point is occupied by a scatterer whose transmission coefficient is with probability p or + with probability 1–p. A perturbation expansion with respect to is derived. The ² term in this expansion shows that the VACF oscillates with time, the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor. The coarse-grained VACF decays for long times like t^–3/2, which is similar to the decay of the VACF of the random Lorentz gas with a single type of scatterer. The perturbation results and the exact ones (found up to 20 collision times) show good agreement.