In order to continue we have to make some assumptions about the conditional averages of the stochastic forces. In view of the chaotic character of the stochastic forces the following assumptions seem to be appropriate

$$ \langle \mathbf{F}(t)\rangle=0, $$

and

$$ \langle \mathbf{F}(t)\cdot \mathbf{F}(t^{\prime })\rangle_{\mathbf{v}_{0}}= C_{\mathbf{v}_{0}}\delta (t-t^{\prime }). $$

We omit the subscript \( \mathbf{v}_{0} \), when the quantity of interest turns out to be independent of \( \mathbf{v}_{0} \). Using the last three equations we get

$$ \langle \mathbf{v}(t)\cdot \mathbf{v}(t)\rangle_{\mathbf{v}_{0}}=v_{0}^{2}e^{-2\xi t}+\frac{C_{\mathbf{v}_{0}}}{2\xi }(1-e^{-2\xi t}). $$

For large t this should be equal to 3kT/m, from which it follows that

$$ \langle \mathbf{F}(t)\cdot \mathbf{F}(t^{\prime })\rangle =6\frac{kT}{m}\xi \delta (t-t^{\prime }). $$

This result is called the fluctuation-dissipation theorem .