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# The Fokker-Planck equation

An alternative way to study stochastic processes is by means of distribution functions . Let be the probability density to find a particle at time t at position with velocity , given that at t=0 it was at position with velocity . We shall demonstrate in Appendix A that this function satisfies

 (4.14)

 (4.15)

where is short hand for . Eq. (4.14) together with the initial value condition Eq. (4.15) is called the Fokker-Planck equation .

We shall not treat this equation any further here, except for one quick remark. The probability to find the particle at time t with velocity , at any position, given it was at t=0 at position with velocity is given by

 (4.16)

where we have assumed the system is translational invariant, i.e. both sides of Eq. (4.16) are independent of . Integrating Eq. (4.14) with respect to we get

 = (4.17) = (4.18)

We may easily obtain the solution to this equation by using the results of the previous section. To this end we notice that according to Eq. (4.4) is a sum of many random vectors. According to the central limit theorem  therefore it must have a Gaussian distribution. The first and second moments are given by

 = 0 (4.19) = (4.20)

where we have used the fluctuation-dissipation theorem  in the form

 (4.21)

Using Eqs. (4.19) and (4.20) we may write

 (4.22)

Although it is a tedious task, it is not difficult to check that this result indeed is the solution to Eqs. (4.17) and (4.18).

Next: The Smoluchowski time scale Up: Stochastic processes Previous: The Langevin equation
W.J. Briels