Mean squared displacement in a generalized Lévy walk model

Abstract

Lévy walks represent a class of stochastic models (space-time coupled continuous time random walks) with applications ranging from the laser cooling to the description of animal motion. The initial model was intended for the description of turbulent dispersion as given by the Richardson's law. The existence of this Richardson's regime in the original model was recently challenged in the work by T. Albers and G. Radons, Phys. Rev. Lett. 120, 104501 (2018): the mean squared displacement (MSD) in this model diverges, i.e. does not exist, in the regime, where it presumably should reproduce the Richardson's law. In the supplemental material to this work the authors present (but do not investigate in detail) a generalized model interpolating between the original one and the Drude-like models known to show no divergences. In the present work we give a detailed investigation of the ensemble MSD in this generalized model, show that the behavior of the MSD in this model is the same (up to prefactiors) as in the original one in the domains where the MSD in the original model does exist, and investigate the conditions under which the MSD in the generalized model does exist or diverges. Both ordinary and aged situations are considered.

References in corpus (6)

• Lévy walks
• Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk
• Rare events and scaling properties in field-induced anomalous dynamics
• Anomalous Drude Model
• Scaling properties of field-induced superdiffusion in Continous Time Random Walks
• Transport in the cat's eye flow on intermediate time scales

Cited by in corpus (9)

• Rare events in generalized Lévy Walks and the Big Jump principle
• Anomalous statistics of laser-cooled atoms in dissipative optical lattices
• Moses, Noah and Joseph Effects in Coupled Lévy Processes
• Infinite invariant density in a semi-Markov process with continuous state variables
• Rare events in stochastic processes with sub-exponential distributions and the Big Jump principle
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Age representation of Levy walks: partial density waves, relaxation and first passage time statistics
• Singularity of Levy walks in the lifted Pomeau-Manneville map
• Complete Decomposition of Anomalous Diffusion in Variable Speed Generalized Lévy Walks