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Consider a particle diffusing in a confined volume which is divided into twoequal regions. In one region the diffusion coefficient is twice the value ofthe diffusion coefficient in the other region. Will the particle spend equalproportions of time in the two regions in the long term? Statistical mechanicswould suggest yes, since the number of accessible states in each region ispresumably the same. However, another line of reasoning suggests that theparticle should spend less time in the region with faster diffusion, since itwill exit that region more quickly. We demonstrate with a simple microscopicmodel system that both predictions are consistent with the information given.Thus, specifying the diffusion rate as a function of position is not enough tocharacterize the behaviour of a system, even assuming the absence of externalforces. We propose an alternative framework for modelling diffusive dynamics inwhich both the diffusion rate and equilibrium probability density for theposition of the particle are specified by the modeller. We introduce anumerical method for simulating dynamics in our framework that samples from theequilibrium probability density exactly and is suitable for discontinuousdiffusion coefficients.

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