Eur. Phys. J. B (2018) 91: 17
https://doi.org/10.1140/epjb/e2017-80401-4

Regular Article

Asymptotic densities from the modified Montroll-Weiss equation for coupled CTRWs^★

¹ Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
² Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel

^a e-mail: ErezAgh5@gmail.com

Received: 6 July 2017
Received in final form: 22 October 2017
Published online: 25 January 2018

Abstract

We examine the bi-scaling behavior of Lévy walks with nonlinear coupling, where χ, the particle displacement during each step, is coupled to the duration of the step, τ, by χ ~ τ^β. An example of such a process is regular Lévy walks, where β = 1. In recent years such processes were shown to be highly useful for analysis of a class of Langevin dynamics, in particular a system of Sisyphus laser-cooled atoms in an optical lattice, where β = 3/2. We discuss the well-known decoupling approximation used to describe the central part of the particles’ position distribution, and use the recently introduced infinite-covariant density approach to study the large fluctuations. Since the density of the step displacements is fat-tailed, the last travel event must be treated with care for the latter. This effect requires a modification of the Montroll-Weiss equation, an equation which has proved important for the analysis of many microscopic models.