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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, defineFor nonzero frequency, with spectral function $\rho (\omega )=-2\mathrm{Im}{G}_R(\omega )$ and $\beta =1/(k_{B}T)$,Here $k_B$ is the Boltzmann constant and $\hslash$ is the reduced Planck constant. In units $k_B=\hslash =1$, this reduces to $S^{>}(\omega )=\rho (\omega )/(1-e^{-\omega /T})$.

To see the origin, the Kubo--Martin--Schwinger condition gives $S^{<}(\omega )=e^{-\beta \hslash \omega }{S}^{>}(\omega )$ for the reversed ordering. The commutator spectral function obeys $\hslash \rho =S^{>}-S^{<}$, which proves the displayed relation. The symmetrized spectrum instead satisfiesAt $|\hslash \omega |\ll k_{B}T$, both forms approach $k_{B}T\rho (\omega )/\omega$. This connects a measured noise spectrum with the dissipative part of the corresponding response, after matching the experimental spectrum's ordering and Fourier conventions. ^[ref]

Chapter 3 of Kinetic Theory by David Tong. Sections 3.1–3.2, printed pages 53–64: Langevin dynamics, Einstein relation (3.12), Gaussian diffusion and the Fokker-Planck equation. Whole-work author credit applies. No separately contributed chapter author is established.