Eur. Phys. J. Special Topics 229, 711–728 (2020)
https://doi.org/10.1140/epjst/e2020-900210-x

Review

Superstatistics and non-Gaussian diffusion

University of Potsdam, Institute for Physics & Astronomy, Karl-Liebknecht-Str 24/25, 14476 Potsdam, Germany

^a e-mail: rmetzler@uni-potsdam.de

Received: 28 September 2019
Received in final form: 22 October 2019
Published online: 12 March 2020

Abstract

Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect “ensembles” of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed (“superstatistical”) transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches.