A paradox of state-dependent diffusion and how to resolve it

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.

Pseudochemotactic drifts of artificial microswimmers

“…[19,20] and numerically investigated in Refs. [21] for thermal Brownian motion in confined geometries. However, the magnitude of the phenomenon reported here is significantly larger and more easily accessible to experimental demonstration.…”

Equilibrium of a Brownian particle with coordinate dependent diffusivity and damping: Generalized Boltzmann distribution

“…In the context of equilibrium, coordinate dependent damping and diffusion are sources of long standing controversy [9][10][11][12][13][14][15]. Brownian motion with coordinate dependent diffusion and damping, in general, is discussed in this paper.…”

Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance

“…Therefore, we arrive at the conventional BT equation (1), where v from Eq. (1) takes the form of [27, 28]…”

“…Substituting it back into the Fokker-Planck equation, we get the Smoluchowski equation: $${\partial }_{t}P=\nabla (D\nabla P)-\nabla (bD\beta \mathit{\mu }P)=\nabla (D(\nabla -\beta \mathbf{F})P),$$ see also in [27, 28]. …”