We shall derive the Fokker-Planck equation by looking at for any function . Because we are always interested in averages like this, equations that may be derived using this object are all we need. (In mathematical terms is a distribution or generalized function, not an ordinary function).

Our proof very much resembles the one in Appendix 3.C. Our starting point is
again the Chapman-Kolmogorov equation

(4.40) |

Multiplying by and integrating yields

Now we shall perform the integral with respect to

(4.42) |

Introducing this into Eq. (4.41) we get

Now we make use of

which hold true to the first order in . The first three of these equations are obvious. The last three easily follow from the Langevin equation in section 4.1 together with the fluctuation-dissipation theorem Eq. (4.21).

= | |||

(4.50) |

Next we change the integration variable into and perform some partial integrations, obtaining

= | |||

(4.51) |

Because this has to hold true for all possible we conclude that the Fokker-Planck equation (4.14) must hold true.