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

摘要

We examine the bi-scaling behavior of Levy walks with nonlinear coupling, where chi, the particle displacement during each step, is coupled to the duration of the step, tau, by chi similar to tau(beta). An example of such a process is regular Levy walks, where beta = 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 beta = 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.