Featured Post

#71 Fundamentals--Langevin equation 常識集_朗之萬方程

We have previously discusses the Langevin equation in episode 14 (http://biophys3min.blogspot.tw/2016/09/14-how-enzymes-propel-themselves-iv.html). Due to its great importance, we will discuss more properties of Langevin equation in this episode.

Review on Episode 14    簡短的回顧

We have previously introduced the Langevin equation, which is basically a stochastic generalization of the Newton's second law:
or written in terms of velocity
in which  F(t) follows

For a one dimensional system and a system satisfying equipartition theorem, the above equation implies
which we have derived in episode 14. However, we skipped the problem of the magnitude of noise by multiplying the equation by x and using the fact that
這我們在第14集有說明過了。不過我們那時候採用了一點偷工減料的方式,把方程式兩邊同乘x, 然後利用下面的事實,規避掉雜訊的大小到底有多大的問題。
Today, we will start from discussing the magnitude of noise.

Magnitude of Noise   雜訊的大小

Let's start from the velocity form of Langevin equation:
Its solution is clear-cut
This equation could be solved by numerical method, or we could switch to the equivalent Fokker-Planck equation. Both of them will be discussed in the following episodes.

Consider the average of square of velocity:

From the definition of white noise, it is easily to show that
Since for large t, the average should be equal to , which means the magnitude of noise should be

Overdamped Langevin Equation  過阻尼朗之萬方程

For a biophysicist, we often consider the overdamped Langevin equation because the system we are interested in (molecules and proteins in water) is a small-Reynold number system in which the inertial effect is negligible. The Langevin equation could then be written in a form with a potential V but without inertial term:
對於生物物理學家而言, 過阻尼版本的朗之萬方程其實比朗之萬方程更常使用,因為我們感興趣的分子、蛋白質是處在一個低雷諾數的環境。在雷諾數低的情況下,流體運動為層流,慣性的效應可以忽略。因此我們拿出原本的朗之萬方程,移除慣性的項之後,在原本的隨機力之外,額外加上一個保守力,我們就寫出了過阻尼朗之萬方程:
It is important to recognize that this simplification would render the transient dynamics of the system incorrectly.

The overdamped Langevin equation would give rise to the Einstein relation if we take  
如果我們把保守力的項移除,那過阻尼朗之萬方程就會變成random walk model,而我們就可以得到前面已經推過好幾次的Einstein relation:
The overdamped Langevin equation would be used heavily in the following episodes.

Bialek, W. 2012. Biophysics: Searching for Principles. The US: Princeton University Press.