Complementary Densities of Lévy Walks: Typical and Rare Fluctuations

A. Rebenshtok ¹   ; S. Denisov ^{2 ,  3 ,  4}   ; P. Hänggi ²   ; E. Barkai ¹

¹ Department of Physics, Institute of Nanotechnology and Advanced Materials Bar-Ilan University, Ramat-Gan, 52900, Israel
² Institute of Physics, University of Augsburg, Universitätsstrasse 1 D-86135, Augsburg Germany
³ Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine
⁴ Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia

Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 76-106

Résumé

Strong anomalous diffusion is a recurring phenomenon in many fields, ranging from the spreading of cold atoms in optical lattices to transport processes in living cells. For such processes the scaling of the moments follows 〈|x(t)|q〉 ∼ tqν(q) and is characterized by a bi-linear spectrum of the scaling exponents, qν(q). Here we analyze Lévy walks, with power law distributed times of flight ψ(τ) ∼ τ−(1+α), demonstrating sharp bi-linear scaling. Previously we showed that for α > 1 the asymptotic behavior is characterized by two complementary densities corresponding to the bi-scaling of the moments of x(t). The first density is the expected generalized central limit theorem which is responsible for the low-order moments 0 q α. The second one, a non-normalizable density (also called infinite density) is formed by rare fluctuations and determines the time evolution of higher-order moments. Here we use the Faà di Bruno formalism to derive the moments of sub-ballistic super-diffusive Lévy walks and then apply the Mellin transform technique to derive exact expressions for their infinite densities. We find a uniform approximation for the density of particles using Lévy distribution for typical fluctuations and the infinite density for the rare ones. For ballistic Lévy walks 0 α 1 we obtain mono-scaling behavior which is quantified.

Détail

DOI : 10.1051/mmnp/201611306

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