From the definitions of $$R(t)$$ and $$\zeta(t)$$, it is straightforward to show that there is a relation between them of the form
$\langle R(0)R(t)\rangle = kT\zeta(t)$
This relation is known as the second fluctuation dissipation theorem. The fact that it involves a simple autocorrelation function of the random force is particular to the harmonic bath model. We will see later that a more general form of this relation exists, valid for a general bath. This relation must be kept in mind when introducing models for $$R(t)$$ and \zeta(t). In effect, it acts as a constraint on the possible ways in which one can model the random force and friction kernel.