= Fluctuation-dissipation theorem
{wiki}
The <fluctuation-dissipation theorem> relates equilibrium fluctuations to dissipative <linear response>. The ordering and <normalization> of a quantum <correlation function> matter. For a <Hermitian operator> $A$ in <thermal equilibrium>, define
$$
S^>(\omega)=\int_{\mathbb R}dt\,e^{i\omega t}\langle A(t)A(0)\rangle,\qquad
G_R(t)=-\frac{i}{\hbar}\Theta(t)\langle[A(t),A(0)]\rangle.
$$
For nonzero <frequency>, with <spectral function> $\rho(\omega)=-2\operatorname{Im}G_R(\omega)$ and $\beta=1/(k_BT)$,
$$
S^>(\omega)=\frac{\hbar\rho(\omega)}{1-e^{-\beta\hbar\omega}}.
$$
Here $k_B$ is the <Boltzmann constant> and $\hbar$ is the <reduced Planck constant>. In units $k_B=\hbar=1$, this reduces to $S^>(\omega)=\rho(\omega)/(1-e^{-\omega/T})$.
To see the origin, the <Kubo--Martin--Schwinger condition> gives $S^<(\omega)=e^{-\beta\hbar\omega}S^>(\omega)$ for the reversed ordering. The <commutator> <spectral function> obeys $\hbar\rho=S^>-S^<$, which proves the displayed relation. The symmetrized spectrum instead satisfies
$$
S_{\rm sym}(\omega)=\frac{\hbar}{2}\coth\left(\frac{\beta\hbar\omega}{2}\right)\rho(\omega).
$$
At $|\hbar\omega|\ll k_BT$, both forms approach $k_BT\rho(\omega)/\omega$. This connects a measured noise spectrum with the dissipative part of the corresponding response, after matching the experimental spectrum's ordering and Fourier conventions. <Kinetic Theory by David Tong>[Section 4.3.2, printed pp. 96–98, equations 4.34–4.37, ordered correlator and thermal reordering; hbar and spectral-function normalization declared here]{ref}
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