Featured Post

#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.

Derivation of Fokker-Planck equation  


Consider the Langevin equation in its general form:
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:
in which represent the Gaussian noise with unit variance.

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
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

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:
And that is the Fokker-Planck equation.

In the following 2 episodes we will talk about the application of Fokker-Planck equation. So please stay tuned!!