Reaction Spreading in Systems With Anomalous Diffusion

TL;DR

This paper reviews anomalous diffusion, especially strong anomalous diffusion, and explores its effects on reactive systems, including front propagation and reaction spreading on complex structures like graphs.

Contribution

It highlights the impact of strong anomalous diffusion on reactive systems and emphasizes the role of probability distribution tails and graph connectivity in reaction spreading.

Findings

Strong anomalous diffusion exhibits non-Gaussian distributions.

The shape of the probability tail influences front propagation speed.

Connectivity dimension affects reaction spreading on graphs.

Abstract

We briefly review some aspects of the anomalous diffusion, and its relevance in reactive systems. In particular we consider {\it strong anomalous} diffusion characterized by the moment behaviour $\langle x(t)^q \rangle \sim t^{q \nu(q)}$, where $\nu(q)$ is a non constant function, and we discuss its consequences. Even in the apparently simple case $\nu(2)=1/2$, strong anomalous diffusion may correspond to non trivial features, such as non Gaussian probability distribution and peculiar scaling of large order moments. When a reactive term is added to a normal diffusion process, one has a propagating front with a constant velocity. The presence of anomalous diffusion by itself does not guarantee a changing in the front propagation scenario; a key factor to select linear in time or faster front propagation has been identified in the shape of the probability distribution tail in absence of reaction. In addition, we discuss the reaction spreading on graphs, underlying the major role of the connectivity properties of these structures, characterized by the {\em connectivity dimension}