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

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

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

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

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

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).