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Anomalous is ubiquitous. (English) Zbl 1238.60086

Authors’ abstract: Brownian motion is widely considered the quintessential model of diffusion processes – the most elemental random transport processes in science and engineering. Examples of diffusion processes displaying highly non-Brownian statistics – commonly called “anomalous diffusion” processes – are omnipresent both in the natural sciences and in engineering systems. The scientific interest in anomalous diffusion and its applications has been growing fast in 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 \(1/f\) 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.

MSC:

60J60 Diffusion processes
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