Influence of external potentials on heterogeneous diffusion processes

TL;DR

This paper analytically examines how external power-law forces influence heterogeneous diffusion processes, revealing that such forces modify the diffusion coefficient but do not alter the fundamental anomalous diffusion exponent.

Contribution

It provides explicit analytical solutions for transition probabilities in heterogeneous diffusion under external power-law forces, expanding understanding of anomalous diffusion behavior.

Findings

External forces change the anomalous diffusion coefficient.

The diffusion exponent remains unaffected by external forces.

Power-law external forces can limit the duration of anomalous diffusion.

Abstract

In this paper we consider heterogeneous diffusion processes with the power-law dependence of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous diffusion. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on the behavior of the diffusion coefficient. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the position. We obtain analytic expressions for the transition probability in two cases: when the power-law exponent in the external force is equal to 2eta-1, where 2eta is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force has a linear dependence on the position. We found that the power-law exponent in the dependence of the mean square displacement on time does not depend on the external force, this force changes only the anomalous diffusion coefficient. In addition, the external force having the power-law exponent different from 2eta-1 limits the time interval where the anomalous diffusion occurs. We expect that the results obtained in this paper may be relevant for a more complete understanding of anomalous diffusion processes.