     Next: Monomer motion Up: The Rouse chain Previous: Normal mode analysis

# Correlation of the end-to-end vector

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 k values which are extremely small compared to N. We therefore write (6.36)

where the prime at the summation sign indicates that only terms with odd kshould occur in the sum. Then (6.37)

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

From Eqs. (6.32) to (6.33) we get = (6.38) = (6.39)

In Appendix A of this chapter it is shown that (6.40)

Introducing everything into Eq. (6.37) we get (6.41)

The characteristic decay time at large t is , which is proportional to N2.

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 t and leads to the desired result.     Next: Monomer motion Up: The Rouse chain Previous: Normal mode analysis
W.J. Briels