Subdiffusive Transport in Heterogeneous Patchy Environments

TL;DR

This paper develops a model for subdiffusive transport in heterogeneous patchy environments with tempering effects, revealing long-term equilibrium behavior distinct from untempered systems where particles accumulate in the slowest patches.

Contribution

The study introduces a novel model incorporating tempering into anomalous transport, showing how it affects long-term particle distribution across patches.

Findings

Long-time equilibrium with constant transport rates

Tempering prevents particle accumulation in slow patches

Distinct behavior from untempered subdiffusive systems

Abstract

Transport across heterogeneous, patchy environments is a ubiquitous phenomenon spanning fields of study including ecological movement, intracellular transport and regions of specialised function in a cell. These regions or patches may be highly heterogeneous in their properties, and often exhibit anomalous behaviour (resulting from e.g. crowding or viscoelastic effects) which necessitates the inclusion of non-Markovian dynamics in their study. However, many such processes are also subject to an internal self-regulating or tempering process due to concurrent competing functions being carried out. In this work we develop a model for anomalous transport across a heterogeneous, patchy environment subject to tempering. We show that in the long-time an equilibrium may be reached with constant effective transport rates between the patches. This result is qualitatively different from untempered systems where subdiffusion results in the long-time accumulation of all particles in the patch with lowest anomalous exponent, $0<\mu<1$.