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Derivation of a Fokker-Planck equation from a Langevin equation

+ 4 like - 0 dislike
333 views

Please tell me how to derive from the Fokker-Planck equation for an open system with the chemical potential the corresponding Langevin equation.

@André_1 The Fokker-Planck equation for N particles is:

\(\partial_t P+\{ H;P \}-\lambda \partial_\vec v \vec v P = D \partial_\vec v^2P \)


The corresponding Langevin equation (\(\vec L(x,v,V_{noise} |) =0\)) is:

\(\partial_t \vec x=\{H;\vec v \}\)

\(\partial_t \vec v=\{H;\vec x \}- \lambda \vec v +\vec f_{noise}\)


If we include the chemical potential:

\(\partial_t P+\{ H;P \}-\lambda \ ( \partial_\vec v \vec v\ -N\mu ) P = D \partial_\vec v^2P \)

then what form will the deformed Langevin equation take?

\(\vec L_{\mu}(x,v,V_{noise} |) =0\)

What is \(\vec L_{\mu}\) ?

asked Oct 25, 2016 in Theoretical Physics by andrey [ revision history ]
edited Oct 26, 2016 by Arnold Neumaier
Most voted comments show all comments

To be understable, you should be more specific and also add a reference to the context.

As I see it, it is necessary to deform the Hamilton or Poisson brackets function. From the statement of the problem should be how to do it.

Someone this issue has already been comprehended.

Who is it?

The relation between Langevin equations and Fokker-Planck equations is discussed in many places. For your case, see, e.g., https://en.wikipedia.org/wiki/Fokker–Planck_equation#Many_dimensions

The equation in the form proposed above does not keep the probabilities norms !

The equation in the form proposed in

https://en.wikipedia.org/wiki/Fokker-Planck_equation#Many_dimensions

preserve normal probability.

A similar situation exists in quantum mechanics and optics: 

\(i \partial_t U=\hat H U\)

\(i \partial_t U=\hat H U-i\alpha U\)

In the second case, the absorption does not give the amount of be maintained substances.

Association with the classics of quantum mechanics described in the literature in both cases

Your proposal is faulty if probability is not conserved.

Any well-designed stochastic process must preserve the total probability. If necessary you need to include explicitly a sink to ensure that.

Most recent comments show all comments

''The intensity is a measure of the probability to detect a photon.''  -  No; it is a measure of the rate of photon detection events, not of a probability.

Probabilities always sum to 1!

I do not have access to the full version of this article, unfortunately.
However, in the abstract, I did not see a problem Langevin and Fokker Planck Association.

In the process of searching, I found an article(

Generalized Fokker-Planck equation: Derivation and exact solutions

DOI: 10.1140/epjb/e2009-00126-3

)  

in a formal form of the equation is present with varying measure of probability in law of linear dissipation  (so when (eq 29) ).  

So,if 

\(q\to 0\) or \(1/g\to 0\)

@ak 

However, it seems to me highly questionable.

 

 

1 Answer

+ 2 like - 0 dislike

For what you want to do, the simple names Langevin equation and Fokker-Plack equation are misleading and no longer justified. For attempts to create a theory of probability non-conserving generalized Langevin equations and Fokker-Plack equations see, e.g., 

Pollak, E. and Berezhkovskii, A.M., 1993. Fokker–Planck equation for nonlinear stochastic dynamics in the presence of space and time dependent friction. The Journal of chemical physics, 99(2), pp.1344-1346.

Berezhkovskii, A. M., Yu A. D’yakov, and V. Yu Zitserman. "Smoluchowski equation with a sink term: Analytical solutions for the rate constant and their numerical test." The Journal of chemical physics 109, no. 11 (1998): 4182-4189.

answered Oct 29, 2016 by Arnold Neumaier (12,375 points) [ no revision ]

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