#73 The equivalence of Langevin equation and Fokker-Planck equation 福克-普朗克方程與朗之萬方程

After discussing the Langevin equation and its computer simulation, today we are going to introduce a completely different approach to the stochastic dynamics. Using Taylor expansion, we will show that a stochastic ODE could actually be transformed into a deterministic PDE. The corresponding PDE of Langevin equation is called Fokker-Planck equation.
在討論了朗之萬方程和他的電腦模擬之後，我們今天要來介紹另一種處理隨機微分方程的方式。利用對朗之萬方程進行泰勒展開，我們可以把具有隨機性的一個常微分方程轉換成一個不具有隨機性的偏微分方程，而這個偏微分方程我們就稱之為福克-普朗克方程。

Derivation of Fokker-Planck equation  

福克-普朗克方程的推導

Consider the Langevin equation in its general form:
考慮一個朗之萬方程如下：

in which η is a white noise with unit variance.
其中 η 是一個變異數為1的白雜訊，這我們在前幾集都已經討論過很多次。

In episode 72 (http://biophys3min.blogspot.tw/2016/09/72-computer-simulation-of-langevin.html), we have talked about the discretization of Langevin equation:
而在上一集中我們也討論了如何把朗之萬方程寫成離散的形式：

in which represent the Gaussian noise with unit variance.
而在離散化的過程中，白雜訊會被高斯雜訊取代。

Now consider another function F = F(x(t)), from the above discretization, we could write down the discretization of F:
現在假設另一個函數F = F(x(t)), 根據上面的離散表示，我們可以寫出F的離散化表示為：

The Taylor expansion tells us that
因此利用泰勒展開，我們就可以得到像這樣的東西

Now if we re-divide our equation with dt and take the average, the Gaussian noise would cancel out and the above equation could be simplified as
把上面的算式除以dt，然後對算式兩邊取平均，利用高斯雜訊的性質，我們就可以把原本的方程式簡化成像下面的形式：

Due to the stochastic nature, obtaining the definite position of the system is impossible. However, the probability distribution of position of the system should be derivable. Since we have expressed our equation in terms of the average, which could be easily expressed in terms of the integral of probability distribution. So let's assume that we describe our stochastic system in terms of probability distribution P(x,t). By definition of the average, we could write down
因為我們的系統本質上是隨機的，因此要知道每個時刻的確切位置是不可能的事情。但是這個系統的位置的機率分布應該是可以獲得的。既然我們已經把原本的方程式用平均的方式來表達，那只要把他寫成機率分布的積分，我們就可以知道機率分布的統御方程了。因此假設我們的系統位置的機率分布為 P(x,t), 根據平均的定義：

By equating both side of the above equation, we now have our governing equation for the unknown probability distribution. However, the integral equation is generally difficult to solve. So here is the most tricky part. Consider a special function of x(t), the delta function:
所以只要把前面的方程式裡的平均代換一下，我們就可以得到機率分布的統御方程式了。然而這是一個積分方程，通常非常難解，所以這裡要使出一個小花招，我們假設F(x)是一個delta function:

Substitute it back to our equation, we will get: (with the aids of integration by parts)
把delta function帶回去積分方程，利用delta function的性質和一點點分部積分，我們可以得到：

And we find that the original stochastic ODE becomes a PDE of the probability distribution:
再代回去之後我們就發現，原本的隨機常微分方程式現在變成機率分布的偏微分方程式了：

And that is the Fokker-Planck equation.
而這就是福克-普朗克方程。


In the following 2 episodes we will talk about the application of Fokker-Planck equation. So please stay tuned!!
在接下來的兩集裡面，我們會討論福克-普朗克方程的應用，所以各位敬請期待囉！