# What is the proper way to model diffusion in inhomogeneous media (Fokker-Planck or Fick's law) and why?

I'm quite confused with the following problem. Normally a one-dimensional Fokker-Planck equation is written in the following form:

$$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\psi)+\frac{\partial^2}{\partial x^2}(D\psi)$$

While traditional convection-diffusion equation without sources has the form:

$$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\psi)+\frac{\partial}{\partial x}(D\frac{\partial \psi}{\partial x})$$

Considering non-constant diffusion $D=D(x,t)$ these equations significantly differ, that looks surprising, because they should interchangeably fit to the same problems (e.g. here). Is there any profound reason/physical explanation for such difference?

Or more straightforwardly: both equations are supposed to describe the evolution of $\psi$ with given $F(x,t)$ and $D(x,t)$. Suppose I have my distribution of something $\psi$ and corresponding coefficients, how can I then decide what form of equation I should use?

P.S. When one writes down a Langevin equation for Brownian motion with non-constant diffusion there appears a so-called noise-induced drift term and the corresponding Fokker-Planck equation then has a form of convection-diffusion equation that I referred earlier.. meaning that "classical" F-P equation is then suitable only for the constant diffusion, which is totally incorrect.. eventually I got lost completely.

• $\partial_x^2\left(D\psi\right)\neq\partial_x\left(D\partial_x\psi\right)$ for $D\neq\text{const}$, so it should not at all be surprising that they differ (cf. this answer of mine) – Kyle Kanos Sep 13 '17 at 16:11
• @KyleKanos To my understanding both equations should be the same since you can use them for the same problems (they both F-P equations or vice versa), yet they differ, normally you write differently F-P and convectional-diffusion equations. I don't get what physics is behind that difference. – funnyp0ny Sep 13 '17 at 16:25
• You can only use them both for the same problem provided $D=\text{const}$ in both cases. If $D\neq\text{const}$, then they are clearly not the same and cannot be used for the same problems. The physics is the diffusion coefficient is now spatially dependent, so $\partial_xD\neq0$; aside from that, I'm not sure what the question really is here. – Kyle Kanos Sep 13 '17 at 16:33
• @KyleKanos Ok, let's say it this way. Both equations are supposed to describe the evolution of $\psi$ with given $F(x,t)$ and $D(x,t)$. Suppose I have my distribution of something $\psi$ and corresponding coefficients, how can I then decide what form of equation I should use? – funnyp0ny Sep 13 '17 at 16:54
• This is a good question, though formulated differently you may get a better response; "is it correct to use Fokker-Planck or Fick's law to model diffusion in inhomogeneous media". The answer is not immediately obvious to me, though I would intuitively prefer the former as a natural starting point on the basis that it is derived (from the Master equation) and the latter is an empirical equation, which maybe you can derive from the former with a further approximation (the formulations are only off by a term). – alarge Sep 16 '17 at 20:11

It is a sticky question, and as van Kampen puts it, " no universal form of the diffusion equation exists, but each system has to be studied individually." https://link.springer.com/article/10.1007/BF01304217 (Unfortunately, I don't have full access to his paper, but you might be able to get it through your library.)

Now, the main reason the question is sticky is that it exposes an ambiguity in the Langevin description. In the Wikipedia article you link to, it says that an Itô process whose Langevin equation reads $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t,$$ then the respective Fokker-Planck equation is $$\frac{\partial{p}}{\partial{t}}=-\frac{\partial{\left[\mu p\right]}}{\partial x} +\frac{1}{2}\frac{\partial^2\left[\sigma^2p\right]}{\partial x^2}$$ where $\sigma^2/2=D$.

Notice that they distinguished that it is an Itô process. If it had been a Stratonovich process, i.e. $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)\circ dW_t,$$ the Fokker-Planck equation would read $$\frac{\partial{p}}{\partial{t}}=-\frac{\partial{\left[\mu p\right]}}{\partial x} +\frac{1}{2}\frac{\partial}{\partial x}\left[\sigma\frac{\partial} {\partial x}\left(\sigma p\right)\right].$$ So now there are two different Fokker-Planck equations in addition to Fick's second law? What gives?

The issue is that when you write down the Langevin process, having $\sigma$ have a spatial dependence causes the noise term to have a non-linear influence on the position. In the Ito picture, the noise is treated as if it were kicking the Brownian particle at the beginning of each time interval $\Delta t$. In the Stratanovich convention, the noise is averaged between the endpoints of the time interval. Depending on whether you integrate using the Stratanovich convention or the Ito one, you get different results. There is also another convention called the Isothermal convention, and this gives a Fokker-Planck equation that looks a bit closer to Fick's Law. Here are a few references, which you should be able to access: http://www.bgu.ac.il/~ofarago/shakedthesis.pdf and https://arxiv.org/pdf/1402.4598.pdf

Cheers!

• Thanks a lot, great answer! And thanks for the references as well. Somehow I always considered both conventions fully equivalent. But still it is rather a technical reason. If two formalisms lead to two different realities, one would expect some fundamental feature of the process, indicating how it should be treated then. Looking through some papers after your answer I've found funny conclusion, in short "Ito is mathematically correct, while Stratonovich describes what actually happens in nature". – funnyp0ny Oct 27 '17 at 9:39

For completeness and for future reference I wish to add a bit to the answer of @AlbertB, in particular to add the following references -

In short, modeling a random process usually starts with writing an appropriate Langevin equations which describe the local microscopical dynamics. These equation include a stochastic (random) variables, and solved per particular realization of the noise. When the noise term is multiplicative (state dependent), that is, the magnitude of the noise term is related to the state of the system - the solution requires what is called an interpretation. Different interpretations have different physical meanings, which manifest in different physical systems, and usually have very different solutions. The most popular interpretations are Ito, Stratanovich and Hanggi-Klimontovich. Following a procedure that averages the noise over trajectories and generates a proper Fokker-Planck equation for the specified Langevin equation - different interpretations result in different Fokker-Planck equations. Reference 1 includes the proper versions of Langevin equations and their appropriate Fokker-Planck equations for the three popular interpretations. Reference 2 compares three physical systems which require different interpretations to make sense.

• Many thanks for the clarification and the great articles! – funnyp0ny Dec 20 '18 at 8:57