Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity

TL;DR

This paper analytically and numerically investigates anomalous diffusion in systems where the diffusion coefficient varies with concentration, revealing conditions under which diffusion deviates from classical behavior.

Contribution

It provides a theoretical framework and numerical verification for anomalous diffusion caused by concentration-dependent diffusivity, highlighting specific initial conditions affecting diffusion behavior.

Findings

Anomalous diffusion occurs with power-law concentration dependence.

Initial delta function leads to anomalous diffusion under certain exponents.

Step initial conditions result in classical square root time spreading.

Abstract

We show analytically that there is anomalous diffusion when the diffusion constant depends on the concentration as a power law with a positive exponent or a negative exponent with absolute value less than one and the initial condition is a delta function in the concentration. On the other hand, when the initial concentration profile is a step, the profile spreads as the square root of time. We verify our results numerically using particles moving stochastically.