The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

TLDR

Fractional dynamics has matured into a robust stochastic framework, with a surge of research papers extending and applying it since the 2000 review on the fractional Fokker–Planck equation. This review aims to synthesize these recent developments into a cohesive perspective. It surveys theoretical modeling of sub‑ and superdiffusive processes—especially superdiffusion—alongside applications such as boundary‑value formulations for first‑passage time densities, and discusses anomalous dynamics across nanoscale, biological, geophysical, and environmental systems. Reference: Rep.

Abstract

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.