In this appendix we derive the Smoluchowski equation directly from the Fokker-Planck equation . This appendix will be rather technical.

In a potential field the Fokker-Planck equation reads

On the Smoluchowski timescale we are interested in

= | (4.53) | ||

= | (4.54) |

We first notice that also satisfies Eq. (4.52), but with initial value condition

For large values of we expect that the velocities will continually be in thermal equilibrium, i.e. that the equilibration time of the velocities is small on the Brownian timescale. We therefore expect that

(4.56) |

From Eq. (4.55) we see that this certainly holds true in the limit of

We now define the projection operator *P* by

(4.57) |

where may be any function of . We write the Fokker-Planck equation as

The projection can easily be shown to have the following properties

P^{2} |
= | P |
(4.61) |

= | 0 | (4.62) | |

= | 0 | (4.63) | |

= | 0 | (4.64) |

We shall now derive the equation of motion of . The Smoluchowski equation, i.e. the equation of motion of then follows at once from

Defining

= | (4.65) | ||

= | (4.66) |

we get from Eq. (4.58)

We now solve Eq. (4.68) using , and introduce the result in Eq. (4.67):

This result is exact, and therefore not very useful.

We shall now simplify Eq. (4.69) by means of some
approximations based on physical grounds. First, since we want to describe
the Smoluchowski timescale, we let
be large, obtaining

(4.70) |

Next we invoke the Markov property; the coordinates develop slowly, meaning that the system quickly loses its memory of the past, and can only depend on the value of at time

It is a simple but rather tedious task to prove that

(4.72) |

Using this in Eq. (4.71) yields

= | |||

= | |||

= | (4.73) |

where in the last step we have used the overdamped condition . Introducing the definition of

(4.74) |

which is recognized as the Smoluchowski equation after using .