## 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.

Original language | English |
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Pages (from-to) | 76-106 |

Number of pages | 31 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Publisher Copyright:© 2016 EDP Sciences.

## Keywords

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