= Solution
The <fluctuation-dissipation theorem> relates spontaneous fluctuations in <thermal equilibrium> to the dissipative part of the <linear response> to a weak conjugate force. For an overdamped trapped coordinate, the <Langevin equation> is
$$
\gamma\dot x+Kx=f(t)+\xi(t),\qquad \langle\xi(t)\xi(t')\rangle=2\gamma k_BT\delta(t-t').
$$
The <fluctuation-dissipation relation for a Langevin particle> fixes the noise strength from the same drag coefficient $\gamma$ that dissipates motion. With a drive proportional to $e^{-i\omega t}$, the <linear response> function is $\chi(\omega)=1/(K-i\gamma\omega)$. The two-sided angular-frequency <power spectral density> then obeys
$$
\boxed{S_{xx}(\omega)=\frac{2\gamma k_BT}{K^2+\gamma^2\omega^2}=\frac{2k_BT}{\omega}\operatorname{Im}\chi(\omega).}
$$
Its integral $\int S_{xx}\,d\omega/(2\pi)=k_BT/K$ agrees with the <equipartition theorem>. Without a restoring force, the same noise-drag relation gives the diffusion-mobility relation $D=k_BT/\gamma$.
In <thermal calibration of an optical trap>, record a bead's spontaneous displacement fluctuations and compare their spectrum with its response to a weak calibrated oscillatory force. The <overdamped harmonic-trap power spectrum> and the dissipative response must be consistent with the same temperature and drag. The equilibrium and weak-drive conditions are essential: an actively driven bath need not obey this equilibrium relation.
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