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.

21 pages, 6 figures. Second round of revisions. This is the version that will appear in Proc Roy Soc

References in corpus (4)

• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Influence of Noise on Force Measurements
• Critical dynamics of ballistic and Brownian particles in a heterogeneous environment
• Deterministic Diffusion in Periodic Billiard Models

Cited by in corpus (11)

• Pseudo-Chemotactic Drifts of Artificial Microswimmers
• Langevin dynamics in inhomogeneous media: Re-examining the Itô-Stratonovich Dilemma
• Brownian motion with multiplicative noises revisited
• Interplay of fast and slow dynamics in rare transition pathways: the disk-to-slab transition in the 2d Ising model
• Metropolis Integration Schemes for Self-Adjoint Diffusions
• Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media
• Generalization of Stokes-Einstein relation to coordinate dependent damping and diffusivity: An apparent conflict
• Crystallization and flow in active patch systems
• Multiphase Partitions of Lattice Random Walks
• Self-organization without heat: the geometric ratchet effect
• Enhanced Diffusion and Chemotaxis of Enzymes