Stochastic one-dimensional Lorentz gas on a lattice

We study a one-dimensional stochastic Lorentz gas where a light particle mo ves in a fixed array of nonidentical random scatterers arranged in a lattic e. Each scatterer is characterized by a random transmission/reflection coef ficient. We consider the case when the transmission coefficients of the sca tterers are independent identically distributed random variables. A symboli c program is presented which generates the exact velocity autocorrelation f unction (VACF) in terms of the moments of the transmission coefficients. Th e VACF is found for different types of disorder for times up to 20 collisio n 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. Th en a lattice point is occupied by a scatterer whose transmission coefficien t is eta with probability p or eta+epsilon with probability l-p. A perturba tion expansion with respect to epsilon is derived. The epsilon(2) term in t his expansion shows that the VACF oscillates with time, the period of oscil lation being twice the time of flight from one scatterer to its nearest nei ghbor. 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 t ype of scatterer. The perturbation results and the exact ones (found up to 20 collision times) show good agreement.