When should we use the Langevin equation and when should use Fokker-Planck

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• Sciencestd
In summary: The Fokker-Planck equation is used to calculate the variance of a process described by a Langevin equation.
Sciencestd
TL;DR Summary
I can't understand the main difference and the main use why I should use this and not that and vice versa and when I should do that.
As everyone knows that we can go from Langevin equation to Fokker-Planck equation which gives the evolution of probability density function. But what I don't understand is what is exactly the main difference between them as long as they are both give the variance (which then we can for example calculate the particle's size). I can't understand the main difference and the main use why I should use this and not that and vice versa and when I should do that.

The question is what you want to do. The Langevin equation describes, e.g., Brownian motion of a particle in a medium as a stochastic process. If you want to simulate it on a computer you use it with some random-number generator for the stochastic force with a given correlation function (or a ##\delta## function if you have the usual case of white noise and no memory effect, i.e., a Markovian process).

If you want to calculate the phase-space probability distribution as a function of time, you solve the Fokker-Planck equation with the appropriate initial condition. This then describes the process in terms of probability theory.

Sciencestd and DrClaude
Thank you so much for your answer! Does it say anything regarding the variance of both? And let's say a particle stuck in optical evanescent field, does the variance will have different value if I got it from fokker-planck equation or from Langevin equation?

It's not clear to me what you mean by the question about a particle "stuck in optical evanescent field", but the statistical properties of the motion of particles subject to random forces in terms of a Langevin equation are described by the corresponding Fokker-Planck equation derived from it. The purpose of the Fokker-Planck equation is to describe the statistical properties of the motion descibed by the Langevin equation in terms of the phase-space distribution function.

Sciencestd
I think I can understand basically the purposes of the two concepts.

What I meant about Stuck in optical evanescent field is...Well let's say we have such in this article: https://pubs.acs.org/doi/abs/10.1021/acsphotonics.6b00628
The Langavin's equation for a particle trapped in gradient force due to evanescent field on the wavegiude:
ma=F(drag)+F(trap)+W(white noise), we assume that ma (the inertia force) is so small comparing to F(drag)
Then we have: F(drag)=F(trap)+W(white noise)
Then they went to Fokker Planck equation.. calucalate the variance.. what is confused me why not just to solve Langevin equation to get the variance instead to use Fokker-Planck equation (or so called smoluchowski euqation, because they wanted the position of the particle and not the velocity)...
I just wanted to give you a picture what is confusing me.

vanhees71

1. When should we use the Langevin equation?

The Langevin equation should be used when studying the dynamics of a single particle in a stochastic system. It is commonly used in fields such as statistical mechanics and molecular dynamics to describe the motion of particles under the influence of random forces.

2. When should we use the Fokker-Planck equation?

The Fokker-Planck equation should be used when studying the evolution of a probability distribution in a stochastic system. It is commonly used in fields such as finance and biology to model the behavior of complex systems with many interacting particles.

3. What is the difference between the Langevin and Fokker-Planck equations?

The Langevin equation describes the motion of a single particle in a stochastic system, while the Fokker-Planck equation describes the evolution of a probability distribution in a stochastic system. The Langevin equation is a stochastic differential equation, while the Fokker-Planck equation is a partial differential equation.

4. Can the Langevin and Fokker-Planck equations be used interchangeably?

No, the Langevin and Fokker-Planck equations cannot be used interchangeably. They describe different aspects of stochastic systems and are derived from different principles. However, under certain conditions, the two equations can be related to each other through the fluctuation-dissipation theorem.

5. Which equation is more suitable for studying biological systems?

The Fokker-Planck equation is more suitable for studying biological systems as it takes into account the interactions between multiple particles and the evolution of a probability distribution. This is important in biological systems where many particles are constantly interacting and influencing each other's behavior.

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