Talk:Fokker–Planck equation

Untitled

How to solve the F-P equation is the question.

I removed the following part because it doesn't make sense. the equation operates on a state space, not on observables. No reason to introduce macrovariables.

of position and velocity of a particle, but it can be generalized to any other observable, too. It applies to systems that can be described by a small number of "macrovariables", where other parameters vary so rapidly with time that they can be treated as noise.

Relationship with stochastic differential equations

In the following equation:

$D_{ij}^{2}(\mathbf {x} ,t)={\frac {1}{2}}\sum _{k}\sigma _{ik}(\mathbf {x} ,t)\sigma _{kj}^{\mathsf {T}}(\mathbf {x} ,t).$

$\sigma _{kj}^{\mathsf {T}}(\mathbf {x} ,t)$ is presumably a scalar, so why the transpose? Shouldn't it just be $\sigma _{jk}(\mathbf {x} ,t)$?

The meaning of $\sigma _{kj}^{\mathsf {T}}(\mathbf {x} ,t)$ is $(\sigma ^{\mathsf {T}})_{kj}(\mathbf {x} ,t)=\sigma _{jk}(\mathbf {x} ,t)$. Either way, the previous article page was wrong (it was $\sigma _{ik}\sigma _{kj}$ instead of $\sigma _{ik}\sigma _{jk}$) —Preceding unsigned comment added by 63.85.81.39 (talk) 21:48, 27 May 2009 (UTC)

Computational considerations

Text says "..instead of this computationally intensive approach, one can use the Fokker–Planck equation..", but is the following information below relevant/new & can appropriate text be put in by someone qualified to do so? (Is it a non-computationally intensive approach, aswell as not being necessary to use the Fokker–Planck equation?) It is from a citation for Prof W.T.Coffey (author of a book on The Langevin Equation) when he became a Fellow of the American Physical Society in 1999: "Coffey, William Thomas [1999] Trinity College Dublin. Citation: For development of new methods for the solution of the nonlinear Langevin equation without the use of the Fokker-Planck equation, allowing the exact calculation of correlation times and mean first passage times." —Preceding unsigned comment added by 193.113.57.161 (talk) 15:55, 29 April 2011 (UTC)

Notes, references, books

Currently, footnotes are lumped together with references. This looks very bad aesthetically. And then there is a section on books. I propose to introduce a "formal" footnotes section and then to merge the books with the other references. --Михал Орела (talk) 10:40, 10 December 2009 (UTC)

Sounds good. Plastikspork _―Œ^(talk) 16:55, 10 December 2009 (UTC)

Formulation

Casually perusing the article -- I noticed that the formulation isn't what I expected. For example, in Karatzas and Shreve on p. 282, the F-P is given by,

${\frac {\partial }{\partial t}}\Gamma (t;x,y)={\mathcal {A}}^{\star }\Gamma (t;x,y),$
$({\mathcal {A}}^{\star }f)(y)={\frac {1}{2}}\sum _{i=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}}{\partial y_{i}\partial y_{k}}}[a_{ik}(y)f(y)]-\sum _{i=1}^{d}{\frac {\partial }{\partial y_{i}}}[b_{i}(y)f(y)].$
where $dX_{t}=b(t,X_{t})dt+\sigma (t,X_{t})dW_{t}$ and $a_{ik}(t,x)=\sum _{j=1}^{r}\sigma _{ij}\sigma _{kj}(t,x)$.  

See also the formulation in Kolmogorov backward equations (diffusion). It appears conventional to include a factor of ${\frac {1}{2}}$ in the definition of the PDE (as opposed to including it in the diffusion term of $D_{2}$ as in the present article. Because considerable work has gone into putting the present formulation, could someone comment on why the particular convention was chosen? I suspect that the convention chosen to facilitate the path-integral formulation -- but I don't have a copy of Zinn-Justin handy.

Consequently, the first formula should probably be modified to $dX_{t}=D_{1}(X_{t},t)dt+{\sqrt {{\color {red}2\cdot }D_{2}(X_{t},t)}}dW_{t}$. Aoschweiger (talk) 11:56, 23 November 2011 (UTC)

This typo still exists in the article. Almost two months later ... I'm going to go ahead and add Aoschweiger's suggestion, since I generally don't appreciate glaring mistakes in wiki pages. IMO, there's no reason to ever introduce the functions D_1 and D_2, since they only introduce confusion. I would lead with the multi-dimensional Ito equation with mu and sigma as the drift and diffusion and go from there (ala the first comment). If no one comments in the next couple of months, I'll make those changes. bradweir (talk) 21:18, 9 January 2012 (UTC)

I have the impression that the first formulation is incorrect: if the process is given by $dX_{t}=D_{1}(X_{t},t)dt+{\sqrt {2\cdot D_{2}(X_{t},t)}}dW_{t}$ then the correct form of the F-P equation should be ${\frac {\partial }{\partial t}}f(x,t)=-{\frac {\partial }{\partial x}}\left[D_{1}(x,t)f(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D_{2}(x,t)f(x,t)\right].$ Marco aita (talk) 23:13, 4 March 2012 (UTC)

Confusing sentence in the introduction

I find the following sentence very confusing:

"is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper."

It seems to imply that Kolmogorov was the person who first introduced the concept which I don't think is the case. Shouldn't it say that Kolmogorov discovered the concept independently in 1931 instead? Sprlzrd (talk) 20:47, 20 April 2015 (UTC)

I agree. The references currently include papers by Fokker and Planck from 1913 and 1917. Also the article on the Kolmogorov backward equations (diffusion) currently says that "Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation." I will change the sentence in the introduction. PeR (talk) 07:37, 19 October 2016 (UTC)

1D vs. Many Dimensions Inconsistency

The 1D equation

${\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial }{\partial x}}D(x,t){\frac {\partial }{\partial x}}p(x,t)$

seems inconsistent with the one for many dimensions

${\frac {\partial p(\mathbf {x} ,t)}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[\mu _{i}(\mathbf {x} ,t)p(\mathbf {x} ,t)\right]+{\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}\left[D_{ij}(\mathbf {x} ,t)p(\mathbf {x} ,t)\right]$

I believe the 1D one should be

${\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}[D(x,t)p(x,t)]$

Any comments?

Juliusbier (talk) 13:49, 17 October 2017 (UTC)

Yes, it's wrong. The coefficient of the randomness should be next to the probability, i.e. it gets differentiated twice. This is rather important to get right! I haven't edited the page though. This is correct:

${\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}[D(x,t)p(x,t)]$ — Preceding unsigned comment added by 86.174.37.127 (talk) 09:10, 18 October 2017 (UTC)

Zero-diffusion limit

should be the Continuity equation, not Liouville's theorem, which involves advective derivative. kiwi (talk) 16:52, 9 July 2021 (UTC)