As an example of a dynamic characteristic of the Rouse chain we calculate
,
i.e. the time correlation
function of the end-to-end vector . First we notice that

(6.35) |

Our result will be dominated by

(6.36) |

where the prime at the summation sign indicates that only terms with odd

where we have used the fact that different modes are uncorrelated.

From Eqs. (6.32) to (6.33) we get

In Appendix A of this chapter it is shown that

Introducing everything into Eq. (6.37) we get

(6.41) |

The characteristic decay time at large

Notice that in this derivation we have averaged over all initial values. We
might also have calculated
,
i.e. the time correlation function of
,
given some initial configurations of the chain. For very large *t* and
the result should be independent of
.
Indeed using
Eq. (6.38) twice, it is not difficult to find

(6.42) |

which for very large