Non-anomalous diffusion is not always Gaussian

TL;DR

This paper demonstrates that standard mean square displacement growth does not necessarily imply Gaussian diffusion, revealing complex behaviors such as multiscaling and non-Gaussian distributions in certain random walk models.

Contribution

It uncovers scenarios where linear MSD growth coexists with non-Gaussian features, challenging the assumption that Gaussianity is implied by standard diffusion.

Findings

High-order moments and probability densities can show multiscaling despite linear MSD.

Certain fractal graphs exhibit standard diffusion without Gaussian distributions.

Standard scaling of moments does not guarantee Gaussian probability densities.

Abstract

Through the analysis of unbiased random walks on fractal trees and continuous time random walks, we show that even if a process is characterized by a mean square displacement (MSD) growing linearly with time (standard behaviour) its diffusion properties can be not trivial. In particular, we show that the following scenarios are consistent with a linear increase of MSD with time: i) the high-order moments, $\langle [x(t)]^q \rangle$ for $q>2$ and the probability density of the process exhibit multiscaling; ii) the random walk on certain fractal graphs, with non integer spectral dimension, can display a fully standard diffusion; iii) positive order moments satisfying standard scaling do not imply an exact scaling property of the probability density.