Fluctuation–dissipation theorem (FDT)

Identifies linear response (dissipation) with equilibrium fluctuations via time-correlation functions; classical and quantum forms.

Fluctuation–dissipation theorem (FDT)

Setup (linear response)

Consider an equilibrium system perturbed by a weak time-dependent field $h(t)$ coupled to an observable $B$ :

H \;\mapsto\; H - h(t)\,B.

For another observable $A$ , the linear response is written

\delta\langle A(t)\rangle = \int_{-\infty}^{t} R_{AB}(t-s)\,h(s)\,ds,

where $R_{AB}(t)$ is the (causal) response kernel and $\langle\cdot\rangle$ denotes an equilibrium ensemble average .

Classical FDT (common time-domain form)

Under standard assumptions of equilibrium and time-translation invariance (often in the canonical ensemble ), one form of the FDT relates the response to an equilibrium correlation:

R_{AB}(t) = \beta\,\theta(t)\,\langle \dot{B}(0)\,A(t)\rangle_{\mathrm{eq}},

where $\beta = 1/(k_B T)$ , $\theta(t)$ is the Heaviside step function (causality), and the correlation is a two-point function .

Equivalent variants differ by integration by parts or by the precise choice of conjugate variables (e.g., $R_{AB}(t)= -\beta\,\theta(t)\,\frac{d}{dt}\langle B(0)A(t)\rangle_{\mathrm{eq}}$ when boundary terms vanish).

Quantum FDT (frequency-domain form)

In quantum equilibrium (a Gibbs state satisfying the KMS condition ), let $\chi_{AB}(\omega)$ be the susceptibility (Fourier transform of the causal response) and let $S_{AB}(\omega)$ be the Fourier transform of an equilibrium correlation (appropriately symmetrized, depending on convention). A standard statement is that

S_{AB}(\omega)\quad\text{is proportional to}\quad \frac{1}{1-e^{-\beta\hbar\omega}}\,\Im \chi_{AB}(\omega),

so the imaginary part of the response (dissipation) is fixed by equilibrium fluctuations.

Consequences

Green–Kubo: Time-integrated equilibrium current correlations yield transport coefficients; see Green–Kubo relations .
Equilibrium input: The FDT expresses a nonequilibrium response purely using equilibrium objects: temperature (temperature ), correlations, and the unperturbed dynamics.
Microscopic reversibility: In stochastic settings, detailed balance is a common sufficient condition behind the equilibrium structure used by FDT.

Link to Kubo’s formula

A systematic derivation of $R_{AB}$ in both classical and quantum settings is given by the Kubo formula .