L\'evy walks and scaling in quenched disordered media

TL;DR

This paper investigates how Le9vy walks behave in one-dimensional disordered media with long-tailed scatterer distributions, revealing different asymptotic behaviors depending on averaging procedures and confirming findings with simulations.

Contribution

It provides a detailed analysis of Le9vy walks in quenched disordered media, deriving asymptotic behaviors and highlighting differences in averaging methods, supported by numerical simulations.

Findings

Asymptotic mean square displacement depends on scatterer distribution exponent.

Different averaging procedures can lead to different asymptotic behaviors.

Numerical simulations confirm theoretical predictions.

Abstract

We study L\'evy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites, in analogy with recent experiments. Our results are compared with numerical simulations, with excellent agreement.