In this section we shall investigate the Gaussian chain at a level where the distance between two consecutive beads may be considered to be small.

Suppose we study a Gaussian chain in an external field with potential
.
The distribution in configuration space then reads

where

where and . Notice that the Green's function is not normalized. One easily verifies however

This equation is called the Chapman-Kolmogorov equation . For this equation to hold true, it is essential that the second summation in the exponent in Eq. (3.13) starts at

Now we shall treat *N* like a continuous variable. In Appendix C we shall
prove that
is the solution of

Accepting this for the moment we may immediately write down the formal solution

One easily checks this by introducing Eq. (3.17) into Eq. ( 3.15) and using Eq. (3.18). Eq. (3.16) is nothing but the closure equation of the complete set of functions.

Knowing
makes it possible to calculate all
kinds of averages:

(3.19) |

(3.20) |

etc.