Truncated Levy Random Walks and Generalized Cauchy Processes

TL;DR

This paper introduces a continuous Markovian model for truncated Levy random walks that incorporates nonlinear friction and velocity-dependent noise saturation, revealing distinct motion regimes and a generalized Cauchy distribution for particle displacement.

Contribution

It extends previous models by including nonlinear effects and provides numerical analysis demonstrating different motion regimes and the distribution of displacements.

Findings

Identification of three motion regimes: ballistic, Levy type, and Brownian.

Displacement distribution in the Levy regime follows a generalized Cauchy form with cutoff.

Particle motion properties depend mainly on the ratio of friction coefficient to noise intensity.

Abstract

A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.