Complementary Densities of Lévy Walks: Typical and Rare Fluctuations

A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai

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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.

Original languageEnglish
Pages (from-to)76-106
Number of pages31
JournalMathematical Modelling of Natural Phenomena
Issue number3
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016 EDP Sciences.


  • Bi-fractal
  • Infinite densities
  • Large deviations
  • Lévy walks
  • Strong anomalous diffusion
  • Superdiffusion


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