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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.
在74和75集中,我們要來討論福克-普朗克方程的應用。在這集裡我們會先用福克-普朗克方程來解穩定狀態的機率分布,而在下集中我們會來解首次通過時間點問題。

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

Recall that for a Langevin equation
回憶之前我們提過,對一朗之萬方程
It's associated Fokker-Planck equation would be
我們都能找到他對應的福克-普朗克方程
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
因為福克-普朗克方程描述的是機率分布的偏微分方程,而機率的時間變化必須和機率流有關。因此我們可以把福克-普朗克方程的右手邊寫成機率流的形式:
As the probability flux becomes constant 0, the probability distribution becomes stationary and it is the steady state solution of Fokker-Planck equation.
只要機率流變成在處處都等於0,機率分布就不會再改變,而這就是福克-普朗克方程的穩定狀態解。

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:
在第72集中我們使用一個雙位能阱作為朗之萬方程電腦模擬的演示:
 The overdamped Langevin equation of this system could be written as
在這個系統中,過阻尼朗之萬方程可以寫成像這樣子
 And thus its equivalent Fokker-Planck equation would be
而他對應的福克-普朗克方程就可以寫成

To find its steady state solution, the problem reduces to the ODE of solving probability flux:
要解出機率分布的穩定狀態解,我們可以把上面的偏微分方程化成以機率流做表達的常微分方程:
It is easy to solve and the solution is
而他的解很容易,就像這個樣子
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:
其中常數A可以藉由積分與正規化來獲得。如果我們比較電腦模擬的機率分布和解析解,我們發現他們彼此非常吻合:
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!
這個技巧非常的實用,我們也可以利用它來猜測某一特定的分布情形對應到的物理機制為何。希望能對各位有幫助。下一集我們會來看首次通過時間點問題,敬請期待囉!

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