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#74 The application of Fokker-Planck equation-I 福克-普朗克方程的應用-I

In episode 74 & 75, we are going to talk about some practical application of Fokker-Planck equation. In this episode, we will use Fokker-Planck equation to obtain the solution of steady state probability distribution, and we will solve the first passage time problem in the following episode.

Interpreting Fokker-Planck Equation 解讀福克-普朗克方程

Recall that for a Langevin equation
It's associated Fokker-Planck equation would be
Since the Fokker-Planck equation describes a PDE of probability distribution, the time evolution of probability should be associated with probability flux. So we could rewrite the RHS of the equation as
As the probability flux becomes constant 0, the probability distribution becomes stationary and it is the steady state solution of Fokker-Planck equation.

An Example 一個簡單的例子

In episode 72 (http://biophys3min.blogspot.tw/2016/09/72-computer-simulation-of-langevin.html), we used a double potential well system as a demonstration of computer simulation of Langevin equation:
 The overdamped Langevin equation of this system could be written as
 And thus its equivalent Fokker-Planck equation would be

To find its steady state solution, the problem reduces to the ODE of solving probability flux:
It is easy to solve and the solution is
The constant A could be obtained by integration and normalization. If we compare our simulation result and the analytical solution, we will find that they match perfectly with each other:
This technique is very helpful when it comes to guessing an underlying physical mechanism associated with a particular distribution pattern. We will talk about the first passage time problem next. Stay tuned!