Scattering lengths and universality in superdiffusive L\'evy materials

TL;DR

This paper investigates how scattering lengths influence superdiffusive Levy walks in disordered and fractal quasi-lattices, revealing a universal superscaling behavior independent of scattering length values.

Contribution

It introduces a scaling framework showing scattering length effects can be absorbed into a multiplicative coefficient, leading to universal scaling functions in Levy walks.

Findings

Scattering length effects are reabsorbed into a multiplicative coefficient.

Scaling exponents and functions are independent of scattering length.

Analytic expressions match numerical simulations with high accuracy.

Abstract

We study the effects of scattering lengths on L\'evy walks in quenched one-dimensional random and fractal quasi-lattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk probability distribution, we show that the effect of the varying scattering length can be reabsorbed in the multiplicative coefficient of the scaling length. This leads to a superscaling behavior, where the dynamical exponents and also the scaling functions do not depend on the value of the scattering length. Within the scaling framework, we obtain an exact expression for the multiplicative coefficient as a function of the scattering length both in the annealed and in the quenched random and fractal cases. Our analytic results are compared with numerical simulations, with excellent agreement, and are supposed to hold also in higher dimensions