= Generalized fluctuation-dissipation relation
{title2=$\mathbb E[R(t)R(t-u)]=\Theta\kappa(u)$}
For the equilibrium <generalized Langevin equation> with unit mass, the random force has zero <mean> and the displayed <autocorrelation>, where $\Theta$ is the equilibrium <velocity> <variance>. If the deterministic relaxation $g$ is stable, $g(0)=1$ and $g'=-\kappa*g$, then the stationary <velocity autocorrelation function> is $\chi=\Theta g$, equivalently
$$
\widetilde\chi(s)=\frac{\Theta}{s+\widetilde\kappa(s)}.
$$
One proof uses the <power spectral density>. The even force <autocorrelation> has spectrum $2\Theta\operatorname{Re}\widetilde\kappa(i\omega)$. With transfer function $H(i\omega)=[i\omega+\widetilde\kappa(i\omega)]^{-1}$, the <velocity> spectrum is $2\Theta|H|^2\operatorname{Re}\widetilde\kappa=2\Theta\operatorname{Re}H$, the spectrum of the even extension of $\Theta g$. Stability removes initial transients with finite second <moment>. A valid force <autocorrelation> must have nonnegative <power spectral density>; an arbitrary proposed <memory kernel> does not automatically satisfy this condition.
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