Complete Decomposition of Anomalous Diffusion in Variable Speed Generalized L\'evy Walks

TL;DR

This paper decomposes the anomalous diffusion in Variable Speed Generalized Lévy Walks into fundamental effects, revealing that multiple mechanisms contribute and that the Noah effect can be arbitrarily strong, expanding understanding of anomalous transport.

Contribution

It introduces a comprehensive decomposition of anomalous diffusion in VGLWs into Joseph, Noah, and Moses effects, highlighting the complex interplay and unbounded Noah exponent in this framework.

Findings

Anomalous diffusion results from combined effects, not a single cause.

The Noah exponent in VGLWs can be arbitrarily large.

VGLWs unify various models and reveal richer diffusion behaviors.

Abstract

Variable Speed Generalized L\'{e}vy Walks (VGLWs) are a class of spatio-temporally coupled stochastic processes that unify a broad range of previously studied models within a single parametrized framework. Their dynamics consist of discrete random steps, or flights, during which the walker's speed varies deterministically with both the elapsed time and the total duration of the flight. We investigate the anomalous diffusive behavior of VGLWs and analyze it through decomposition into the three fundamental constitutive effects that capture violations of the Central Limit Theorem (CLT): the Joseph effect, reflecting long-range increment correlations, the Noah effect, arising from heavy-tailed step-size distributions with infinite variance, and the Moses effect, associated with statistical aging and non-stationarity. Our results show that anomalous diffusion in VGLWs is typically generated by a nontrivial combination of all three effects, rather than being attributable to a single mechanism. Strikingly, we find that within the VGLW framework the Noah exponent $L$, which quantifies the strength of the Noah effect, is unbounded from above, revealing a richer and more extreme landscape of anomalous diffusion than in previously studied L\'{e}vy-walk-type models.