Anomalous diffusion and the Moses effect in a model of aging

TL;DR

This paper analyzes the causes of anomalous diffusion in an aging model by decomposing it into Joseph, Noah, and Moses effects, using analytic and numerical methods to quantify their contributions.

Contribution

It introduces a detailed decomposition of anomalous diffusion into three fundamental effects within a chaotic dynamical system model, with analytic calculations and numerical validation.

Findings

All three effects can cause anomalous diffusion.

Scaling exponents for each effect are analytically derived and numerically confirmed.

The Moses effect plays a significant role in experimental anomalous diffusion.

Abstract

We decompose the anomalous diffusive behavior found in a model of aging into its fundamental constitutive causes. The model process is a sum of increments that are iterates of a chaotic dynamical system, the Pomeau-Manneville map. The increments can have long-time correlations, fat-tailed distributions and be non-stationary. Each of these properties can cause anomalous diffusion through what is known as the Joseph, Noah and Moses effects, respectively. The model can have either sub- or super-diffusive behavior, which we find is generally due to a combination of the three effects. Scaling exponents quantifying each of the three constitutive effects are calculated using analytic methods and confirmed with numerical simulations. They are then related to the scaling of the distribution of the process through a scaling relation. Finally, the importance of the Moses effect in the anomalous diffusion of experimental systems is discussed.