One-dimensional stochastic Lévy-Lorentz gas

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 ξᵢ are independent random variables identically distributed according to the probability density function μ(ξ)~ξ^-(1+γ). We show that under certain conditions the mean square displacement of the particle obeys 〈x²(t)〉>~Ct^3-γ for 1<γ<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.