Comb model with slow and ultraslow diffusion

TL;DR

This paper investigates a generalized two-dimensional comb model with various memory kernels, revealing conditions for anomalous and ultraslow diffusion, and provides a CTRW framework linking the model to Wiener processes.

Contribution

It introduces a generalized diffusion equation on a comb structure with different memory kernels and derives the associated probability distributions and mean squared displacements.

Findings

Anomalous diffusion occurs along both axes.

Ultraslow diffusion is observed under certain kernels.

The CTRW model links the diffusion process to Wiener processes.

Abstract

We consider a generalised diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyse the probability distribution functions and we derive the mean squared displacement in $x$ and $y$ directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both $x$ and $y$ directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.