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
of particles
which were at position
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

where

Using this in Eq. (4.29) while setting , leads to the Einstein equation (4.13).

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

we get the Smoluchowski equation

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

= | (4.34) | ||

= | (4.35) |

Averaging the Fokker-Planck equation over the initial velocities and integrating over , we find the continuity equation Eq. (4.31), with

where the second step serves to define the velocity at time

We next derive an equation describing the time development of the velocity
.
To this end we multiply the Fokker-Planck
equation by ,
average over the initial velocities, and integrate
over ,
obtaining

(4.37) |

Using the continuity equation and rearranging we find

In a strongly damped system the integral on the right hand side yields the velocity fluctuation at position , multiplied by the probability to find the particle at position , which is

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 . Introducing the result into Eq. (4.36) we find the flux Eq. (4.29) and next the Smoluchowski equation. In Eq. (4.39) 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.