A comparison study of slow--subdiffusion and subdiffusion

TL;DR

This paper compares slow-subdiffusion and subdiffusion processes, showing their Green functions can be similar under certain conditions, and suggests slow-subdiffusion can be modeled as subdiffusion with a small alpha and time-dependent diffusion coefficient.

Contribution

It demonstrates that slow-subdiffusion can be effectively described using subdiffusion models with a small alpha and a time-dependent diffusion coefficient.

Findings

Green functions for slow-subdiffusion and subdiffusion can be very similar.

Slow-subdiffusion can be modeled by subdiffusion with a small alpha.

The subdiffusion coefficient D_alpha can depend on time.

Abstract

We study slow-subdiffusion in comparison to subdiffusion. Both of the processes are treated as random walks and can be described within continuous time random walk formalism. However, the probability density of the waiting time of a random walker to take its next step $\omega(t)$ is assumed over a long time limit in the form $\omega(t)\sim 1/t^{\alpha+1}$ for subdiffusion, and in the form $\omega(t)\sim h(t)/t$ for slow-subdiffusion [$h(t)$ is a slowly varying function]. We show that Green functions for slow-subdiffusion and subdiffusion can be very similar when the subdiffusion coefficient $D_\alpha$ depends on time. This creates the possibility of describing slow-subdiffusion by means of subdiffusion with a small value of the subdiffusion parameter $\alpha$, and the subdiffusion coefficient $D_\alpha$ varying over time.