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# Green's function method

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

 (3.12)

where Z is a normalizing constant. In many cases we are interested in properties depending on the position vectors of only a few beads along the chain. It will turn out to be useful then to introduce the Green's function

 (3.13)

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

 (3.14)

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 i=1, and not at i=0 as in Eq. (3.12).

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

 (3.15)

 (3.16)

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

 (3.17)

 (3.18)

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.

Next: One Gaussian chain in Up: The Gaussian chain Previous: The Gaussian chain
W.J. Briels