The Langevin equation

In this chapter we shall study the basics of the theory of stochastic processes . This is most easily done in the language of colloidal suspensions .

Consider a colloidal particle suspended in a liquid. On its path through the liquid it will continuously collide with the liquid molecules. Because on average the particle will collide more often on the front side than on the back side, it will experience a systematic force proportional with its velocity, and directed opposite to its velocity. Besides this systematic force the particle will experience a stochastic force  $\vec{F}(t)$. The equations of motion then read 

 

$$\frac{d\vec{r}}{dt} \vec{v}$$ = (4.1)

$$\frac{d\vec{v}}{dt} -\xi \vec{v}+\vec{F}.$$ = (4.2)

From hydrodynamics  we know that the friction constant  $\xi$ is given by

$$\xi =6\pi \eta a/m$$ (4.3)

where $\eta$ is the viscosity  of the solvent and a is the radius of the particle. We shall derive this and some other results from hydrodynamics in the next chapter.

Solving Eq. (4.2) yields

$$\vec{v}(t)=\vec{v}_{0}e^{-\xi t}+\int_{0}^{t}d\tau e^{-\xi (t-\tau )}\vec{F }(\tau ).$$ (4.4)

If we want to get some useful information out of this, we have to average over all possible realizations of $\vec{F}(t)$, with the initial velocity as a condition. A useful quantity for example is

 

$${\langle \vec{v}(t)\cdot \vec{v}(t)\rangle _{\vec{v} _{0}}=v_{0}^... ...-\xi (2t-\tau )}\vec{v} _{0}\cdot \langle \vec{F}(\tau )\rangle _{\vec{v}_{0}}}$$

$$+\int_{0}^{t}d\tau ^{\prime }\int_{0}^{t}d\tau e^{-\xi (2t-\tau -... ...})}\langle \vec{F}(\tau )\cdot \vec{F}(\tau ^{\prime })\rangle _{ \vec{v}_{0}}.$$ (4.5)

In order to continue we have to make some assumptions about the conditional averages of the stochastic forces. In view of the chaotic character of the stochastic forces the following assumptions seem to be appropriate

  

$$\langle \vec{F}(t)\rangle$$ = 0 (4.6)

$$\langle \vec{F}(t)\cdot \vec{F}(t^{\prime })\rangle _{\vec{v}_{0}} C_{ \vec{v}_{0}}\delta (t-t^{\prime }).$$ = (4.7)

Here and elsewhere we omit the subscript $\vec{v}_{0}$, when the quantity of interest turns out to be independent of $\vec{v}_{0}$. Using Eqs. (4.6) and (4.7) in Eq. (4.5) we get

$$\langle \vec{v}(t)\cdot \vec{v}(t)\rangle _{\vec{v}_{0}}=v_{0}^{2}e^{-2\xi t}+\frac{C_{\vec{v}_{0}}}{2\xi }(1-e^{-2\xi t}).$$ (4.8)

For large t this should be equal to 3kT/m, from which it follows that

$$\langle \vec{F}(t)\cdot \vec{F}(t^{\prime })\rangle =6\frac{kT}{m}\xi \delta (t-t^{\prime }).$$ (4.9)

This result is called the fluctuation-dissipation theorem .

Integrating Eq. (4.4) we get

$$\vec{r}(t)=\vec{r}_{0}+\vec{v}_{0}\frac{1}{\xi }(1-e^{-\xi t}... ...rime }e^{-\xi (\tau -\tau ^{\prime })}\vec{F}(\tau ^{\prime })$$ (4.10)

from which we calculate the mean square displacement 

$$\langle ( \vec{r}(t)-\vec{r}_{0})^{2}\rangle _{\vec{v}_{0}}=\... ...)^{2}+\frac{3kT}{m\xi ^{2}}(2\xi t-3+4e^{-\xi t}-e^{-2\xi t}).$$ (4.11)

For very large t this becomes

$$\langle (\vec{r}(t)-\vec{r}_{0})^{2}\rangle =\frac{6kT}{m\xi }t$$ (4.12)

from which we get the Einstein relation  

$$D= \frac{kT}{m\xi }$$ (4.13)

where we have used $\langle (\vec{r}(t)-\vec{r}_{0})^{2}\rangle =6Dt$.