A friend of mine wrote me privately to ask my opinion on modeling with the Fokker-Planck equation with space-dependent diffusion coefficient. I will reply to him here, given that he also complained that activity on this blog has long been stalling…
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This used to be a widely debated issue: what is the “right” equation, and what does it even mean to be “right”? The argument becomes even more delicate when one goes to Stochastic Differential Equations, where one also has the problem of the interpretation of the stochastic differential, something that drove physicists nuts for generations. I perfectly agree with everything that my friend writes, expecially because he goes back to Van Kampen (that he calls by name, Nico), who was always deep and thoughtful about the foundations.
The problem is that, in phenomenological settings, one never really derives the Fokker-Planck equation from an underlying microscopic model. Rather, its form is assumed on the basis of some desired property. Since Klimontovich (who was obsessed with this issue and managed to introduce his own end-point stochastic differential which I believe is one of the most horrible things ever seen in stochastic processes), it became popular to impose that the steady-state distribution is tMaxwell-Boltzmann. This is perfectly legit as far as 1) the requirement makes sense not ex autoritate, but because there has been some indipendent measurement and/or argument sustaining that property for that particular system (I think all too often the MB distribution is fetishized); 2) the properties tested after modelling and compared to the “theoretical predictions” are indipendent of that particular property that was used to fix the equation. If this is not the case, at best one is just proving that the system is diffusive (which still is a good result).
Such a posteriori* considerations have always been in-built in modeling with stochastic processes. For example, take the derivation of the Einstein relation: on the one side one constructs the correlation function of the Brownian particle from the Langevin equation, on some very basic assumptions about isotropy, homogeneity and decorrelation; on the other side one takes the equipartition theorem, which comes from a quite different approach to the physics of gases (but it would take a perfect historian to understand exactly which idea precedes and which follows, and how independent one is of the other.) In the end, that worked nicely with the Einstein relation, because it allowed to compute Avogadro’s number, which was then confirmed by independent experiments.
Given all these considerations, now that we know the Einstein relation (at least in the homogeneous case), in future modeling, is it part of the assumptions of next diffusion equation or is it part of the consequences? This is a really tricky question, it depends very much on how one formulates the problem. And I can’t take a stance in this…
The next “philosophical” question in line is “what is equilibrium”? Personally I am coming to the conclusion that the obsession with those particular probability distributions that we call “equilibrium” obfuscates our minds. We are putting the cart before the horse. To me, equilibrium is any state such that there are no currents (of the relevant degrees of freedom). Equilibrium of Brownian particles is perfectly compatible with a temperature gradient in the medium: particles in a temperature gradient will arrange themselves in such a way to not flow in any particular direction. This is a different question from what is the reason why the medium has a temperature gradient. Maybe, the “medium behind the medium” is made of heat pumps that maintain the medium in an inhomogeneous state. But now we shifted our attention to other degrees of freedom, and this is a completely different question!
So much so with the philosophical considerations. Going into the technical details, my friend notices that when considering a state-dependent diffusion coefficient the drift is always “effective”. In 1D this fixing the drift to have a prescribed property is pretty straightforward, given that unless one has currents to infinity, the system is always equilibrium (in my sense). But in >1D things become spectacular.
I’ll have to call myself into cause (sorry for the self-promotion…):
MP, Generally covariant state-dependent diffusion, J. Stat. Mech. P07005 (2013), arXiv:1206.2798
The idea is: The diffusion coefficient (or matrix) can be interpreted as a metric on a manifold. Then the question is: is there any interpretation of the stochastic differential (Ito, Stratonovich, “Klimontovich”) that assures that, without effective drift (thus in a situation that we would associated to “equilibrium”, given that there is no forcing…), there won’t be currents flowing through the system? And the answer is no! You always need an effective drift, coming from the geodesic displacement.
So, now, what is your favorite equation, and what are the criteria to build it? There’s no definite answer. We should definitely take nothing for granted, including Klimontovich or Hanggi, and we should make sure that the output of our theory is different than the input…
* By the way, I use post-hoc and a posteriori interchangeably. But thinking about it, I attach a negative feel to the first and a positive feel to the second: maybe because the first hints at a bias that future expectations cast on past observations, while the second hints at un-biasing past expectations after future observations.