Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,731 comments
1,470 users with positive rep
818 active unimported users
More ...

  Derivation of a Fokker-Planck equation from a Langevin equation

+ 4 like - 0 dislike
2440 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 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 \(\phi_\alpha=\phi \sqrt{T_\alpha} \)\(T_\alpha=\int {u(x,t)u(x,t)^*}dx/\int {u(x,0)u(x,0)^*}dx\)

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

_________________________________________________________________________

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.

-----------------------------------------------------------------------------------------------------------------------------------------

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).

''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.

 

 

Most recent comments show all comments

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.

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 

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 (15,787 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar\varnothing$sicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...