Derivation of a Fokker-Planck equation from a Langevin equation

Question

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:

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

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

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

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

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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 \(\vec L_{\mu}\) ?

Comments

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

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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?

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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

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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

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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.

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Schrödinger equation with absorption also does not keep measure "probability".

However, its use in optics as standard.

What prevents to give an equivalent sense of Fokker-Planck equation with absorption and Schrödinger equation with absorption?

The reason for this is simple: the number of particles in the whole space decreases over time.

Let's understand the "probability" as density measures the amount of a substance similar in behavior to the Schrödinger equation(optics).

The question is: how to establish a formal agreement between the Langevin equation and the Fokker-Planck equation with absorption.

I​n the Stochastic quantization a classical  noise and a quantum noise is uniform, so I think the answer is, in some literature.

Although I never found this answer does not.@AF 

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But the number of particles is something different from a probability. Probabilities necessarily sum to one.

In quantum optics, the squared amplitude denotes not a probability but an intensity, which can decay. But stochastic processes are probabilistic and don't make sense when the probabilities don't sum to one. This is the basic assumptions of a probability measure.

What does your Langevin equation model?

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The intensity is a measure of the probability to detect a photon.

How to determine the value of this measure?

There are two natural methods.

a) Normalize the value of the field to the initial value of energy (number of particles)

\( \phi(x,t)=u(x,t) /\sqrt{\int {u(x,0)u(x,0)^*}dx}\)

In this case, the dynamic equations have the form

 \(i \partial_t \phi=\hat H \phi-i \alpha \phi/2\)

b) Normalize the value of the field to the actual value of energy (number of particles):

\( \phi_\alpha (x,t)=u(x,t) /\sqrt{\int {u(x,t)u(x,t)^*}dx}\)

Method (b) guarantee the normalization unit.

In this case, the dynamic equations have the form

 \(i \partial_t \phi_\alpha=\hat H \phi_\alpha \)  

Were \(\phi_\alpha=\phi \sqrt{T_\alpha} \); \(T_\alpha=\int {u(x,t)u(x,t)^*}dx/\int {u(x,0)u(x,0)^*}dx\)

\( \partial_t \ T_\alpha=-\alpha T_\alpha \)

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The choice is determined by the priority calibration simplicity and beauty

Absolutely correct, the term "probability" the right to use only metoda (b)

However, I admit some terminological freedom and use this term in the sense of the first calibration method (a).

Accounting dissipation properties in the Fokker-Planck equation is completely analogous to the Schrödinger equation.

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For me the interesting mathematical formalism, which guarantees an unambiguous association between classical and quantum mechanics (Schrödinger equation).

Similar Containers:

For me the interesting mathematical formalism, which guarantees an unambiguous association between the classic and kinetic (Fokker-Planck equation).

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''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!

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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.

 

 

Answer 1

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.