Equivalence of position-position auto-correlations in the Slicer Map and the L\'evy-Lorentz gas

TL;DR

This paper demonstrates that the position-position auto-correlations in the Slicer Map and the Le9vy-Lorentz gas are equivalent in the asymptotic limit, providing new insights into their diffusive behaviors.

Contribution

It derives analytic expressions for the Slicer Map's correlations and conjectures similar asymptotic behavior for the Le9vy-Lorentz gas, supported by numerical evidence.

Findings

Analytic expressions for Slicer Map correlations

Conjectured asymptotic correlation behavior for Le9vy-Lorentz gas

Numerical agreement with conjectured scaling

Abstract

The Slicer Map is a one-dimensional non-chaotic dynamical system that shows sub-, super-, and normal diffusion as a function of its control parameter. In a recent paper [Salari et al., CHAOS 25, 073113 (2015)] it was found that the moments of the position distributions as the Slicer Map have the same asymptotic behaviour as the L\'evy-Lorentz gas, a random walk on the line in which the scatterers are randomly distributed according to a L\'evy-stable probability distribution. Here we derive analytic expressions for the position-position correlations of the Slicer Map and, on the ground of this result, we formulate some conjectures about the asymptotic behaviour of position-position correlations of the L\'evy-Lorentz gas, for which the information in the literature is minimal. The numerically estimated position-position correlations of the L\'evy-Lorentz show a remarkable agreement with the conjectured asymptotic scaling.