A numerical study of Lévy random walks: Mean square displacement and power-law propagators

Abstract

Non-diffusive transport, for which the particle mean free path grows nonlinearly in time, is envisaged for many space and laboratory plasmas. In particular, superdiffusion, i.e. 〈Δx2〉 ∝tαwith α > 1, can be described in terms of a Lévy random walk, in which case the probability of free-path lengths has power-law tails. Here, we develop a direct numerical simulation to reproduce the Lévy random walk, as distinct from the Lévy flights. This implies that in the free-path probability distributionΨ(x, t) there is a space-time coupling, that is, the free-path length is proportional to the free-path duration. A power-law probability distribution for the free-path duration is assumed, so that the numerical model depends on the power-law slope μ and on the scale distancex0. The numerical model is able to reproduce the expected mean square deviation, which grows in a superdiffusive way, and the expected propagatorP(x, t), which exhibits power-law tails, too. The difference in the power-law slope between the Lévy flights propagator and the Lévy walks propagator is also estimated.