Anomalous is ubiquitous

Brownian motion is widely considered the quintessential model of diffusion processes—the most elemental random transport processes in Science and Engineering. Yet so, examples of diffusion processes displaying highly non-Brownian statistics–commonly termed “Anomalous Diffusion” processes–are omnipresent both in the natural sciences and in engineered systems. The scientific interest in Anomalous Diffusion and its applications is growing exponentially in the recent years. In this Paper we review the key statistics of Anomalous Diffusion processes: sub-diffusion and super-diffusion, long-range dependence and the Joseph effect, Lévy statistics and the Noah effect, and 1/f1/f noise. We further present a theoretical model–generalizing the Einstein–Smoluchowski diffusion model–which provides a unified explanation for the prevalence of Anomalous Diffusion statistics. Our model shows that what is commonly perceived as “anomalous” is in effect ubiquitous.

► The article provides an overview of Anomalous Diffusion (AD) statistics.
► The Einstein–Smoluchowski diffusion model is extended and generalized.
► The generalized model universally generates AD statistics.
► A unified ‘universal macroscopic explanation’ for AD statistics is established.
► AD statistics are shown to be fundamentally connected to robustness.