Continuous Markovian model for Levy random walks with superdiffusive and superballistic regimes

TL;DR

This paper extends a Levy random walk model to include superdiffusive, ballistic, and superballistic regimes, analyzing velocity correlations and Levy statistics crossover through numerical methods.

Contribution

It introduces a numerical analysis of a nonlinear Langevin-based Levy walk model covering multiple dynamical regimes, including superballistic behavior.

Findings

Model describes Levy walks with superdiffusive, ballistic, and superballistic regimes

Velocity correlations decay at specific time scales

Walker displacement exhibits Levy statistics with power-law tails

Abstract

We consider a previously devised model describing Levy random walks (Phys. Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that the given model describes Levy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously only the superdiffusive regime has been analyzed. In this model the walker velocity is governed by a nonlinear Langevin equation. Analyzing the crossover from small to large time scales we find the time scales on which the velocity correlations decay and the walker motion essentially exhibits Levy statistics. Our analysis is based on the analysis of the geometric means of walker displacements and allows us to tackle probability density functions with power-law tails and, correspondingly, divergent moments.