Complementary Densities of Lévy Walks: Typical and Rare Fluctuations

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

^⋆ Corresponding author. E-mail: barkaie@biu.ac.il

Abstract

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⟩ ∼ t^qν(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.