Diffusion in a crowded environment

TL;DR

This paper investigates anomalous diffusion arising from resource depletion in crowded environments, deriving and analyzing macroscopic equations from microscopic models, revealing deviations from classical Fickian diffusion.

Contribution

It introduces a new set of macroscopic diffusion equations derived from microscopic stochastic models, explaining non-power-law anomalous diffusion due to resource depletion.

Findings

Deviations from Fickian diffusion are linked to resource depletion.

Macroscopic equations exhibit non-power-law anomalous diffusion.

Numerical and analytical studies confirm the model's predictions.

Abstract

We analyze a pair of diffusion equations which are derived in the infinite system--size limit from a microscopic, individual--based, stochastic model. Deviations from the conventional Fickian picture are found which ultimately relate to the depletion of resources on which the particles rely. The macroscopic equations are studied both analytically and numerically, and are shown to yield anomalous diffusion which does not follow a power law with time, as is frequently assumed when fitting data for such phenomena. These anomalies are here understood within a consistent dynamical picture which applies to a wide range of physical and biological systems, underlining the need for clearly defined mechanisms which are systematically analyzed to give definite predictions.