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

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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

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:

$\left&space;\langle&space;v(t)v(t)&space;\right&space;\rangle=v(0)v(0)e^{-2\zeta&space;t/m}+2e^{-2\zeta&space;t/m}\int_{0}^{t}\frac{v(0)\left&space;\langle&space;F(t')&space;\right&space;\rangle}{m}e^{\zeta&space;t'/m}\textup{d}t'+e^{-2\zeta&space;t/m}\int_{0}^{t}\textup{d}t''\int_{0}^{t}\textup{d}t'\frac{\left&space;\langle&space;F(t')F(t'')&space;\right&space;\rangle}{m^2}e^{\zeta(t'+t'')/m}$
From the definition of white noise, it is easily to show that

Since for large t, the average should be equal to $k_BT/m$, 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:

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 $\partial&space;V/\partial&space;x=0$

The overdamped Langevin equation would be used heavily in the following episodes.

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