If we introduce the mobility tensors Eq. (5.56) into Eq. (7.7),
we are left with a completely intractable set of equations. One way out of
this is by noting that in equilibrium, on average, the mobility tensors will
be proportional to the unit tensor. We therefore assume that
will
never differ much from
,
and that we may replace the
mobility tensor in Eq. (7.7) by

(7.10) |

where is the Gaussian equilibrium distribution. A simple calculation yields

= | |||

= | |||

= | (7.11) |

The next step is to write down the equations of motion of the normal
coordinates Eq. (6.26):

where

Equation (7.12) still is not tractable. It turns out however (see Appendix), that for large

Introducing this result in Eq. (7.12), we see that the normal modes , just like with the Rouse chain, constitute a set of decoupled coordinates.

For small values of *k* we find

where the first term on the right hand side of Eq. (7.16) equals zero when

(7.18) |

We are now in a position to calculate the diffusion coefficient