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

Part of: Present Achievements in SpacePlasmas

Published online by Cambridge University Press:  16 October 2014

E. M. Trotta  and
G. Zimbardo
E. M. Trotta
Affiliation:
Department of Physics, University of Calabria, Ponte P. Bucci, Cubo 31C, I-87036 Rende, Italy
G. Zimbardo*
Affiliation:
Department of Physics, University of Calabria, Ponte P. Bucci, Cubo 31C, I-87036 Rende, Italy
*
† Email address for correspondence: [email protected]
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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〉 ∝ tα 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 x0. 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.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 1 , January 2015 , 325810108
Copyright
Copyright © Cambridge University Press 2014 

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