Transmission probability through a L\'evy glass and comparison with a L\'evy walk

TL;DR

This study uses simulations to show that superdiffusive light transmission in a L\'evy glass can occur alongside divergent step size moments, challenging the simple L\'evy walk model.

Contribution

The paper demonstrates that superdiffusive transport in L\'evy glasses can coexist with correlated step sizes, providing new insights beyond traditional L\'evy walk assumptions.

Findings

Transmission probability scales as 1/L^{\gamma} with \gamma < 1.

Superdiffusive behavior can occur with divergent second moments of step sizes.

Correlations in step sizes explain deviations from ideal L\'evy walk predictions.

Abstract

Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power law distribution of radii (a socalled L\'evy glass) have found that the transmission probability T \propto 1/L^{\gamma} scales superdiffusively ({\gamma} < 1). The data has been interpreted in terms of a L\'evy walk. We present computer simulations to demonstrate that diffusive scaling ({\gamma} \approx 1) can coexist with a divergent second moment of the step size distribution (p(s) \propto 1/s^(1+{\alpha}) with {\alpha} < 2). This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a L\'evy walk of independent steps.