The Smoluchowski equation

We shall now derive the equivalent of the Fokker-Planck equation, but this time applicable at the Smoluchowski timescale.

Suppose we are given a distribution $G(\vec{r};\vec{r}_{0};t)$ of particles which were at position $\vec{r}_{0}$ at time t=0. We assume that the particles are at every instant of time in thermal equilibrium with respect to their velocities. A flux will exist, given by

$$\vec{\jmath}(\vec{r},\vec{r}_{0},t)=-D\vec{\nabla}G(\vec{r};\... ...ac{1}{m\xi }G(\vec{r};\vec{r}_{0};t)\vec{\nabla}\Phi (\vec{r})$$ (4.29)

where D is the diffusion constant , occurring in $\langle (\vec{r}(t)-\vec{r}_{0})^{2}\rangle =6Dt$, and $\xi$ is the friction coefficient  on the Smoluchowski timescale. At equilibrium, the flux must be zero and the distribution be equal to

$$G_{\mathrm{eq}}( \vec{r})=C\exp \{-\beta \Phi (\vec{r})\}.$$ (4.30)

Using this in Eq. (4.29) while setting $\vec{\jmath}(\vec{r} ,t)=0$, leads to the Einstein equation (4.13).

Introducing Eq. (4.29) into the equation of particle conservation  

$$\frac{\partial }{\partial t}G(\vec{r};\vec{r}_{0};t)=-\vec{\nabla}\cdot \vec{ \jmath}(\vec{r},\vec{r}_{0},t)$$ (4.31)

we get the Smoluchowski equation 

  

$$\frac{\partial }{\partial t}G(\vec{r};\vec{r}_{0};t) \vec{\nabla}\cdot \frac{1}{m\xi }G(\vec{r};\vec{r}_{0};t)\vec{\nabla}\Phi (\vec{r})+\vec{ \nabla}\cdot D\vec{\nabla}G(\vec{r};\vec{r}_{0};t)$$ = (4.32)

$$\lim_{t\rightarrow 0}G(\vec{r};\vec{r}_{0};t) \delta (\vec{r}-\vec{r}_{0})$$ = (4.33)

In the remaining part of this section we shall substantiate the above derivation. First we define the particle distribution on the Smoluchowski timescale by

$$G(\vec{r},\vec{r}_{0};t) \int d^{3}v\bar{G}(\vec{r},\vec{v},\vec{r} _{0};t)$$ = (4.34)

$$\bar{G}(\vec{r},\vec{v},\vec{r}_{0};t) \int d^{3}v_{0}\left\{ \frac{m}{2\pi kT}\right\} ^{\frac{3}{2}}\e... ...1}{ 2}\beta mv_{0}^{2}\}\mathcal{G}(\vec{r},\vec{v},\vec{r}_{0},\vec{v}_{0};t).$$ = (4.35)

Averaging the Fokker-Planck equation  over the initial velocities and integrating over $\vec{v}$, we find the continuity equation Eq. (4.31), with

 

$$\vec{\jmath}(\vec{r},\vec{r}_{0};t) \int d^{3}v\vec{v}\bar{G}(\vec{r}, \vec{v},\vec{r}_{0};t)$$ =

$$G(\vec{r},\vec{r}_{0};t)\vec{V}(\vec{r},\vec{r}_{0};t)$$ = (4.36)

where the second step serves to define the velocity $\vec{V}(\vec{r},\vec{r} _{0};t)$ at time t at position $\vec{r}$, given that the particle was originally at $\vec{r}_{0}$.

We next derive an equation describing the time development of the velocity $\vec{V}(\vec{r},\vec{r} _{0};t)$. To this end we multiply the Fokker-Planck equation by $\vec{v}$, average over the initial velocities, and integrate over $\vec{v}$, obtaining

$$\frac{\partial }{\partial t}G\vec{V}=-\vec{\nabla}\cdot \int ... ...{v} \vec{v}\bar{G}-\xi G\vec{V}-\frac{1}{m}G\vec{\nabla}\Phi .$$ (4.37)

Using the continuity equation  and rearranging we find

$$m\left( \frac{\partial }{\partial t}+\vec{V}\cdot \vec{\nabl... ...a}\cdot \int d^{3}v(\vec{v}- \vec{V})(\vec{v}-\vec{V})\bar{G}.$$ (4.38)

In a strongly damped system the integral on the right hand side yields the velocity fluctuation at position $\vec{r}$, multiplied by the probability to find the particle at position $\vec{r}$, which is GkT1/m. On the left hand side we recognize the acceleration of the particle at $\vec{r}$(see section 5.1). Eq. (4.38) may then be written as

$$m\frac{D}{Dt}\vec{V}=-m\xi \vec{V}-\vec{\nabla}\Phi -kT\vec{\nabla}\ln G.$$ (4.39)

In a strongly damped system the average particle velocity is almost constant. We therefore put the left hand side of Eq. (4.39) equal to zero and solve for $\vec{V}$. Introducing the result into Eq. (4.36) we find the flux Eq. (4.29) and next the Smoluchowski equation. In Eq. (4.39) $-kT\vec{\nabla}\ln G$ is called the Brownian force.

In appendix B we shall present an alternative derivation of the Smoluchowski equation. In appendix C we will demonstrate that the Langevin equations (4.27), (4.28) are equivalent to the Smoluchowski equation.