The fluctuation-dissipation theorem (FDT) is a principle in statistical mechanics that relates the response of a system in thermal equilibrium to small perturbations (dissipation) and the spontaneous fluctuations occurring in the system (fluctuations). In essence, it provides a way to understand how the equilibrium properties of a system influence its dynamics when it is perturbed. The theorem states that the way a system responds to an external force (i.e.
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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 in thermal equilibrium, defineFor nonzero frequency, with spectral function and ,Here is the Boltzmann constant and is the reduced Planck constant. In units , this reduces to .
To see the origin, the Kubo--Martin--Schwinger condition gives for the reversed ordering. The commutator spectral function obeys , which proves the displayed relation. The symmetrized spectrum instead satisfiesAt , both forms approach . 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]
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