Consider a "state-dependent diffusion": a diffusion process for which the diffusion coefficient $D(x)$ depends on the (stochastic) state $x$ of the system. (An example is provided by the diffusion of tracers in a spatially inhomogeneous bath, e.g. with varying viscosity or temperature.) What is the associated flux? If $c(x)$ is the probability to find the system in the state $x$ at a given time, should the flux read $j(x)=-D(x)\nabla c(x)$, as in Fick's law? Or rather $j(x)=-\nabla(Dc)(x)$, as in the Fokker-Planck equation? (I'm using the terms "Fick" and "Fokker-Planck" here only as tags for each alternative.)
It is well known that this question does not have a definite answer, and should be answered case by case. My actual question is threefold:
- Do you know examples of physically relevant state-dependent diffusions? What category do they fall in? Fick, or Fokker-Planck?
- Do you have an intuition for the physics underlying this alternative?
- The Fokker-Planck case has the peculiarity to lead to equilibrium distributions which are not of the Boltzmann-Gibbs form (the steady-state $c(x)$ depends on $D(x)$ and not just on the state's energy). What does this tell us about the foundations of statistical mechanics?