In this section we study the mean square displacements of the individual monomers. Using
Eq. (6.24) and the fact that different modes are not
correlated, we get

(6.43) |

Introducing Eq. (6.38) we get

where we have used Eqs. (6.30) and (6.31) to calculate the first term, and Eqs. (6.39) and (6.40 ) for the second term.

There are two different limits to Eq. (6.44). First, when *t*is very large, i.e.
,
the first term will dominate, yielding

This is consistent with the fact that the polymer as a whole diffuses with diffusion constant

Secondly, suppose
.
Then the sum in Eq. (6.44)
will dominate. Averaging over all monomers, and replacing the sum over *k*by an integral we get

= | |||

= | |||

= | (6.46) |

Performing the final integral we get

So, at short times the mean square displacement of a typical monomer goes like the square root of