A Paradox of State-Dependent Diffusion and How to Resolve It

TL;DR

This paper explores a paradox in state-dependent diffusion where the expected long-term particle distribution contradicts intuitive reasoning, and proposes a new modeling framework that accurately captures the equilibrium behavior.

Contribution

It introduces a novel framework for modeling diffusion that specifies both the diffusion rate and equilibrium density, resolving the paradox and improving simulation accuracy.

Findings

Both predictions about particle distribution are consistent with the model.

The proposed framework accurately captures equilibrium distributions.

A numerical method is developed for simulating the new model.

Abstract

Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients.