One-Dimensional Stochastic Lévy-Lorentz Gas

Abstract

We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers $\{ξ_i \}$ are independent random variables identically distributed according to the probability density function $μ(ξ)\sim ξ^{-(1 + γ)}$. We show that under certain conditions the mean square displacement of the particle obeys $<x^2 (t) > \ge C t^{3 - γ}$ for $1 < γ< 2$. This behavior is compatible with a renewal Lévy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.