### #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.

## 福克-普朗克方程的推導

Consider the Langevin equation in its general form:

in which η is a white noise with unit variance.

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 $\widetilde{\eta}(t)$ 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:

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

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:

Substitute it back to our equation, we will get: (with the aids of integration by parts)

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!!