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 PomeauâManneville 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.
Highlights
According to the Central Limit Theorem, the distribution of a process that is the sum of many random increments will have a variance that grows linearly in time
The root causes of anomalous diffusion can be decomposed into the Joseph, Noah and Moses effects
Nonstationary increments, such as what occurs in aging processes, can cause anomalous diffusion through the Moses effect
Summary
According to the Central Limit Theorem, the distribution of a process that is the sum of many random increments will have a variance that grows linearly in time. For stochastic processes that have stationary increments, that is, increments with a time-independent distribution, Mandelbrot [12] decomposed the nature of anomalous diffusion into two root causes, or, effects. He recognized that it can be caused either by long-time increment correlations or by increment distributions that have sufficiently fat tails, so that their variance is infinite. We decompose the anomalous diffusion found in a simple model of aging behavior [16, 17] and find that it is due to a rich combination of the Joseph, Noah and Moses effects. We discuss our results and the importance of the Moses effect for anomalous diffusive behavior observed in experimental systems
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