One-Dimensional Stochastic L\'evy-Lorentz Gas

Abstract

We introduce a L\'evy-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 $\{\xi_i \}$ are independent random variables identically distributed according to the probability density function $\mu(\xi )\sim \xi^{-(1 + \gamma)}$. We show that under certain conditions the mean square displacement of the particle obeys $<x^2 (t) > \ge C t^{3 - \gamma}$ for $1 < \gamma < 2$. This behavior is compatible with a renewal L\'evy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.