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.