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# Appendix A

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

 (4.41)

Now we shall perform the integral with respect to z on the right hand side. Because differs from zero only when is in the neighbourhood of , we expand around
 (4.42)

Introducing this into Eq. (4.41) we get

 = (4.43)

Now we make use of

 = 1 (4.44) = (4.45) = (4.46) = 0 (4.47) = 0 (4.48) = (4.49)

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). 1 in Eq. (4.49) denotes the 3-dimensional unit matrix. Introducing everything into Eq. (4.43), dividing by and taking the limit , we get
 = (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.

Next: Appendix B Up: Stochastic processes Previous: The Smoluchowski equation
W.J. Briels