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

• Published in: Journal of plasma physics 2015-01, Vol.81 (1), p.np-np, Article 325810108
• Main Authors: Trotta, E. M., Zimbardo, G.
• Format: Article
• Language: English
• Subjects: Coupling; Deviation; Mathematical models; Mean free path; Mean square values; Plasma physics; Present Achievements and New Frontiers in Space Plasmas; Probability distribution; Random walk; Random walk theory; Slopes
• ISSN: 0022-3778; 1469-7807
• DOI: 10.1017/S0022377814000592

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. 〈Δx 2〉 ∝ t α with 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 distribution Ψ(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 distance x 0. The numerical model is able to reproduce the expected mean square deviation, which grows in a superdiffusive way, and the expected propagator P(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.