Stochastic One-Dimensional Lorentz Gas on a Lattice

Abstract

We stud ya one-dimensional stochastic Lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice. Each scatterer is characterized b ya random transmissionreflection 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 functio n( 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 conside ra specific type of disorder :a two-state Lorentz gas in which two types of scatterers are arranged randomly i na lattice. The na lattice point is occupied b ya 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 &32 , which is similar to the decay of the VACF of the random Lorentz gas wit ha single type of scatterer. The perturbation results and the exact one s( found up to 20 collision times) show good agreement.