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.
Answer
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!
No comments:
Post a Comment