State-dependent diffusions: Fick's law vs. Fokker-Planck's, which and why?

Question

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:

1. Do you know examples of physically relevant state-dependent diffusions? What category do they fall in? Fick, or Fokker-Planck?
2. Do you have an intuition for the physics underlying this alternative?
3. 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?

Answer 1

I don't agree with the premise of the question (that there is some mysterious disagreement between hydrodynamics, Fick's law, and the Fokker-Plank equation). I am not entirely certain what you mean by ``state dependent''. I will assume that the system is in local thermodynamic equilibrium. This is the basic assumption in hydrodynamics, and the Fokker-Planck equation is a possible microscopic model of how equilibrium is reached. In this case the diffusion constant is a function of $x$ only through its dependence on thermodynamic variables. In a simple fluid with a single type of impurity these are $T(x)$ and $\mu(x)$.

Note that $j(x)=-D(\mu(x),T(x))\nabla c(x)$ is not the most general statement of Fick's law. In general, there is also thermal diffusion, and there are extra terms in the presence of an external potential. This is explained in standard texts on fluid dynamics (like Landau). The RHS of the diffusion equation is $\nabla [D(\mu(x),T(x))\nabla c(x)]$.

I think that this is indeed the form one obtains from a stochastic model, see for example equ.(312) in Chandrasekhar's review http://rmp.aps.org/abstract/RMP/v15/i1/p1_1 (he calls this the Smoluchowski equation), or equ. (4.19) in these lecture notes http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/LectureNotes/chp4.pdf . There is an extra term in Fick's law, but this term is related to the external potential, $j(x)\sim Dc/T\nabla V$. I also think that this had to be the case. As long as I consider the most general hydro equation any stochastic model that relaxes to local thermodynamic equilibrium should reduce to this equation in the appropriate limit.

Answer 2

I have answers to your first question.

a) One example of the state-dependent diffusion would be the solidification of the melt containing impurities. Assume that you have a melt of one substance and a front of solidification runs forward. The substance, however, has one or several types of impurities that can diffuse. The diffusion factor depends, of course, of the state (solid, or fluid) of the main substance.

b) Another variant of the same story is a situation with a solution of several reagents such that a reaction auto-wave runs in the solvent. It also makes a front. In principle, the species in the solution that do not take part in the auto-wave reaction diffuse differently in the product-poor and product-rich space. This again makes their diffusion to be state-dependent.