In this chapter we shall study the basics of the theory of stochastic processes . This is most easily done in the language of colloidal suspensions .

Consider a colloidal particle suspended in a liquid. On its path through the
liquid it will continuously collide with the liquid molecules. Because on
average the particle will collide more often on the front side than on the
back side, it will experience a systematic force proportional with its
velocity, and directed opposite to its velocity. Besides this systematic
force the particle will experience a stochastic force
.
The equations of motion then read

From hydrodynamics we know that the friction constant is given by

(4.3) |

where is the viscosity of the solvent and

Solving Eq. (4.2) yields

If we want to get some useful information out of this, we have to average over all possible realizations of , with the initial velocity as a condition. A useful quantity for example is

In order to continue we have to make some assumptions about the conditional averages of the stochastic forces. In view of the chaotic character of the stochastic forces the following assumptions seem to be appropriate

Here and elsewhere we omit the subscript , when the quantity of interest turns out to be independent of . Using Eqs. (4.6) and (4.7) in Eq. (4.5) we get

For large

(4.9) |

This result is called the fluctuation-dissipation theorem .

Integrating Eq. (4.4) we get

(4.10) |

from which we calculate the mean square displacement

(4.11) |

For very large

(4.12) |

from which we get the Einstein relation

where we have used .