Universal singularities of anomalous diffusion in the Richardson class

TL;DR

This paper reveals a universal singularity in the cumulant generator of anomalous diffusion processes in inhomogeneous environments, linking non-Gaussian displacement scaling to Richardson class diffusion exponents.

Contribution

It introduces a model demonstrating a universal singularity in the cumulant generator for anomalous diffusion, independent of specific system details.

Findings

Universal singularity in the normalized cumulant generator.

Relation established between asymptotics and Richardson class diffusion exponents.

Numerical tests confirm the theoretical predictions.

Abstract

Inhomogeneous environments are rather ubiquitous in nature, often implying anomalies resulting in deviation from Gaussianity of diffusion processes. While sub- and superdiffusion are usually due to conversing environmental features (hindering or favoring the motion, respectively), they are both observed in systems ranging from the micro- to the cosmological scale. Here we show how a model encompassing sub- and superdiffusion in an inhomogeneous environment exhibits a critical singularity in the normalized generator of the cumulants. The singularity originates directly from the asymptotics of the non-Gaussian scaling function of displacement, which we prove to be independent of other details and hence to retain a universal character. Our analysis, based on the method first applied in [A. L. Stella et al., arXiv:2209.02042 (2022)], further allows to establish a relation between the asympototics and diffusion exponents characteristic of processes in the Richardson class. Extensive numerical tests fully confirm the results.