# Mean squared displacement in a generalized L\'evy walk model

## TL;DR

This paper investigates the mean squared displacement in a generalized Lévy walk model, clarifying when it converges or diverges, and extends understanding of anomalous diffusion in stochastic models.

## Contribution

It provides a detailed analysis of the MSD in a generalized Lévy walk model, identifying conditions for convergence and divergence, and bridging the gap between original and Drude-like models.

## Key findings

- MSD behavior matches original model where it exists
- Conditions for MSD divergence are characterized
- Both ordinary and aged cases are analyzed

## Abstract

L\'evy 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.