Anomalous diffusion and the Moses effect in an aging deterministic model

Abstract

We decompose the anomalous diffusive behavior found in an aging system 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.

Keywords

• anomalous diffusion
• aging
• infinite ergodic theory
• chaos
• nonequilibrium statistical mechanics