We shall now solve the equations (6.3) to (6.6) . We first solve them leaving out the fluctuating
forces. The equations of motion then form a linear set of (3*N*+3) first
order differential equations, whose general solutions are sums of (3*N*+3)independent specific solutions.

As a specific solution we try

The equations of motion then read

where we have used

= | (6.14) |

In order for equations (6.11) to (6.13) to be consistent we need

= | (6.15) | ||

= | (6.16) |

This may be rewritten as

= | (6.17) | ||

= | (6.18) |

We find independent solutions from

a-b |
= | b |
(6.19) |

(N+1)a+b |
= | (6.20) |

So, finally

The specific solution Eq. (6.10), with

In the following we shall frequently make use of

This is evident when

(6.23) |

which is consistent with Eq. (6.22).

We now turn to the solution of Eqs. (6.3) to (6.6
). To this end we write

The factor of two in front of the summation is only for reasons of convenience. Using Eq. (6.22) we may invert this to

The equations of motion then read

Eqs. (6.27) and (6.28) were obtained from Eq. (6.6) and Eq. (6.22) by using

We have finally arrived at a decoupled set of 3(

Using Eq. (6.22) we easily see that
.
Eqs. (6.26) and (6.27)
for *k*=0 read

from which we get again the diffusion coefficient of the centre of mass as given in Eq. (6.9).

The specific solutions that we have found are called the normal modes of the chain. describes the motion of the centre of gravity. The other modes describe vibrations of the chain leaving the centre of mass unchanged.

In the applications ahead of us, our results will always be expressed as
sums over normal modes, which sums will always be dominated by the
contributions from the modes with small *k*, i.e. those with large
wavelength. We therefore approximate Eq. (6.26) by

These equations apply when . The characteristic time is given by