Continuous-time random walk theory of superslow diffusion

TL;DR

This paper investigates superslow diffusion using a continuous-time random walk model, deriving general laws and specific examples for particle spread where variance grows slower than any power of time.

Contribution

It introduces a framework linking superslow diffusion to slowly varying waiting time distributions within the CTRW model, providing general laws and illustrative examples.

Findings

Superslow diffusion occurs with slowly varying waiting time distributions.

Derived general laws for biased and unbiased superslow diffusion.

Illustrated results with specific waiting-time distribution classes.

Abstract

Superslow diffusion, i.e., the long-time diffusion of particles whose mean-square displacement (variance) grows slower than any power of time, is studied in the framework of the decoupled continuous-time random walk model. We show that this behavior of the variance occurs when the complementary cumulative distribution function of waiting times is asymptotically described by a slowly varying function. In this case, we derive a general representation of the laws of superslow diffusion for both biased and unbiased versions of the model and, to illustrate the obtained results, consider two particular classes of waiting-time distributions.