Fluctuation-Dissipation Theorem Overview

Updated 25 December 2025

Fluctuation-Dissipation Theorem is a core principle linking spontaneous equilibrium fluctuations to linear responses under external perturbations.
It underpins classical and quantum formulations (e.g., Kubo formulas, Einstein relations) that quantify transport coefficients and noise phenomena.
Recent generalizations extend FDT to non-equilibrium, non-Gaussian systems and machine learning models, offering insights into effective temperatures and violation mechanisms.

The fluctuation-dissipation theorem (FDT) is a fundamental principle in statistical physics establishing a quantitative relation between the spontaneous fluctuations of a system at thermal equilibrium and its response to weak external perturbations. The FDT is central to linear response theory, forming the basis for understanding relaxation, noise, and transport phenomena in a vast array of physical, chemical, and biological systems. Its classical and quantum formulations underpin key results such as the Einstein relation, Nyquist–Johnson noise, Kubo's linear response formulas, and Green–Kubo relations for transport coefficients. In recent years, the FDT has been extensively generalized to systems far from equilibrium, open quantum systems, machine learning dynamics, and stochastic thermodynamics, including precise formalism for violation, extension, and experimental testing.

1. Classical and Quantum Formulations of the FDT

The FDT relates equilibrium fluctuations of a system observable to its linear response following a small external perturbation. In its classical form, for a dynamical variable $A(t)$ in equilibrium at temperature $T$ , the Langevin approach yields

$m\,\frac{dv}{dt} = -m\gamma v(t) + f(t),$

with Gaussian white noise $\langle f(t) f(t') \rangle = 2m \gamma k_B T \delta(t-t')$ , and the linear susceptibility $\chi(\omega)$ connected to the power spectrum $S_A(\omega)$ via

$S_A(\omega) = 2k_B T\,\text{Im} \chi(\omega)/\omega.$

Onsager's regression hypothesis and Kubo's linear response theory generalize this by formalizing the link between equilibrium autocorrelations and relaxation under perturbation: $\chi(t) = -\frac{1}{k_B T} \frac{d}{dt} \langle A(t)A(0) \rangle,$ and, in the frequency domain, the Green–Kubo relation for transport coefficients.

In quantum systems, the FDT is expressed using commutators and symmetric (Kubo–Martin–Schwinger) correlation functions. For an observable $\hat{A}$ ,

$S(\omega) = \hbar\,\chi''(\omega)\,\coth\left(\frac{\hbar\omega}{2k_B T}\right),$

where $\chi''(\omega)$ is the imaginary part of the Fourier-transformed retarded susceptibility, and $S(\omega)$ is the symmetrized power spectral density: $S(\omega) = \frac{1}{2}\int dt\,e^{i\omega t} \langle \hat{A}(t)\hat{A}(0) + \hat{A}(0)\hat{A}(t) \rangle.$ This quantum relation encapsulates both thermal and zero-point (quantum vacuum) fluctuations, with the Bose–Einstein factor $\coth(\hbar\omega/2k_B T)$ arising generically from the structure of equilibrium Boltzmann weights (Branchina et al., 2010).

2. Physical Origin and Mapping to Oscillator Baths

Branchina et al. established that for any quantum or classical system whose linear response is described by the FDT, the two-point correlation functions can be exactly mapped (in linear response) onto those of a fictitious bath of independent harmonic oscillators, with coupling coefficients determined by the spectral decomposition of the original system. The ubiquitous $\coth(\beta\hbar\omega/2)$ factor in the FDT is not exclusive to real oscillators but derives formally from Boltzmann weights in the eigenbasis. This mapping underpins widely used phenomenological Caldeira–Leggett–type oscillator-bath models for dissipation in metals, Josephson devices, Kondo systems, and quantum dots (Branchina et al., 2010).

3. Extensions Beyond Equilibrium: Generalization, Breakdown, and Effective Temperatures

3.1. Nonequilibrium Steady States and Housekeeping Flux

Feng and Wang, as well as subsequent works (Feng et al., 2011, Zhang et al., 2021), systematically generalized the FDT for non-equilibrium steady states (NESS) with broken detailed balance. The response of an observable $A$ in a multivariate overdamped Langevin system is decomposed as: $R_A(t-t') = -\langle A(t)\,D^{-1}F(x(t'))\rangle - \langle A(t)\,D^{-1} v^{ss}(x(t'))\rangle,$ where the first term is spontaneous relaxation (equilibrium-like), and the second term is the steady-state 'housekeeping' contribution from the persistent probability flux $v^{ss}$ (curl flux), associated in gauge theory language with a non-trivial curvature. In the detailed-balance limit, the flux vanishes, reducing the response to the conventional FDT (Feng et al., 2011).

This formalism reveals that in far-from-equilibrium quantum Markovian systems, the generalized FDT includes an extra correlation term governed by a quantum curl flux. In detailed balance, this vanishes, but in driven open systems it crucially determines the response, as shown, e.g., in molecular junctions where the optical transmission spectrum's non-reciprocal features encode the nonequilibrium curl flux (Zhang et al., 2021).

3.2. Effective Temperatures and Experimental Tests

For aging, glassy, or weakly driven systems, the FDT is often approximately restored through the introduction of an effective temperature $T_{\rm eff}$ , particularly measurable via velocity or stress fluctuations in colloidal gels and glass-formers. Dual measurements of diffusion coefficient $D$ and mobility $\mu$ yield

$T_\mathrm{eff} = D/\mu k_B,$

directly relating the observed fluctuation and dissipation, with equilibrium signaled by $T_\mathrm{eff} = T$ . Experiments confirm that for rapidly relaxing observables, such as tracer velocity in Laponite gels, the FDT is not violated and $T_{\rm eff} \simeq T_{\rm bath}$ , while slow structural observables can exhibit $T_{\rm eff} > T$ (Colombani et al., 2017).

3.3. Quantum Violations

Quantum systems can manifest large-scale, macroscopic violations of the FDT, even in the linear response regime. Notably, in the Hall conductivity of clean two-dimensional electron gases at low temperatures and high fields, the symmetrized current–current correlator measured experimentally disagrees by up to an order of magnitude with the response predicted by the Kubo formula. The violation parameter diverges as $T\to0$ for strong magnetic fields, requiring separate consideration of dissipative versus dissipationless (antisymmetric) response functions and caution in interpreting quantum noise estimations from linear response (Kubo et al., 2018).

4. FDT under Non-Gaussian and Heavy-Tailed Fluctuations

In systems subject to non-Gaussian (e.g., Lévy) noise, the FDT can be exactly generalized by recognizing that the response kernel is no longer related to simple two-point statistics. Instead, if $X_\gamma(x) = -\partial_{\lambda_\gamma} \ln \rho_{\rm ss}(x;\lambda)|_{\lambda=\lambda_0}$ is the conjugate observable to the perturbed parameter, the appropriate generalized FDT is

$\chi_{A,\gamma}(t) = \frac{d}{dt} \langle A(t) X_\gamma(0) \rangle_0,$

with conjugate variables determined from the stationary non-Gaussian PDF. This applies for arbitrary $\alpha$ -stable noise, with heavy tails requiring non-linear $X_\gamma$ for convergence of moments. The procedure restores fluctuation–response relations even for systems with diverging variances, as in superdiffusive transport, turbulent plasmas, or banknote dispersal (Dybiec et al., 2012, Gomes-Filho et al., 2023).

5. Fluctuation-Dissipation in Growth and Coarse-Grained Dynamics

A "hidden" FDT exists for stochastic growth phenomena such as Edwards–Wilkinson and KPZ interfaces, where the noise intensity $D$ is related to the effective surface tension $\nu$ and saturated roughness $w_s$ : $\langle \xi(x,t)\xi(x',t')\rangle = 2b\,\nu\,w_s^2\,\delta(x-x')\delta(t-t').$ This mirrors the role of noise and dissipation in classic Brownian motion, with the variance (here, the roughness) playing the role of the fluctuating observable. Extensions to correlated noise and higher dimensions introduce further non-universal exponents but retain the underlying fluctuation-dissipation link (Gomes-Filho et al., 2021, Gomes-Filho et al., 2023).

In stochastic linear learning models, the FDT also describes parameter fluctuations arising from stochastic gradient descent: $\Sigma_{xx}\,\Sigma_{WW} + \Sigma_{WW}\,\Sigma_{xx} = 2 D,$ where $\Sigma_{xx}$ is the input covariance, and $D$ quantifies the noise amplitude. This result demonstrates the universality of FDT-type relations in high-dimensional data-driven systems (Han et al., 2021).

6. Generalized Langevin Equation and Second FDT

The generalized (Mori–Zwanzig) projection approach yields the generalized Langevin equation (GLE), in which the second FDT ensures that the memory kernel and the autocorrelation of the fluctuating force are related. The GLE admits an exact expression for the noise correlation in terms of the memory kernel, even in nonequilibrium: $\langle \eta(t)\eta(t') \rangle_{\rm ss} = k_B T_{\rm neq} \Gamma(|t-t'|),$ with $T_{\rm neq}$ dependent on the mode, the spatial profile of dissipation, and the local temperature. For non-equilibrium steady states, the correct fluctuation-dissipation balance only emerges when the noise term is constructed to be orthogonal to the relevant slow variables—a property enforced by the deterministic Volterra equation governing the evolution of correlation functions (Zhu et al., 2021, Jung et al., 2021, Fontana et al., 2021).

7. Applications, Experimental Verification, and Open Problems

FDT and its extensions underlie analyses of electrical noise in conductors, parametrizing spectral densities in terms of the real part of the impedance and accounting for quantum corrections, Casimir forces, and the ultraviolet divergence of zero-point fluctuations (Reggiani et al., 2018). Modern approaches extend FDT to resource-theoretic quantum thermodynamic protocols, where the minimal dissipation in, e.g., work extraction or information erasure, is governed by second-order fluctuations of free energy (Biswas et al., 2021).

Direct Perrin-type experimental tests establish the FDT to high accuracy in colloidal gels for velocity observables (Colombani et al., 2017), while recent work probes generalized FDTs in turbulent flows by combining data-driven model reduction (e.g., via DMD modes) and FDT-based ROM construction (Khodkar et al., 2018).

State-of-the-art generalizations apply to nonequilibrium quantum fluids with imposed temperature gradients, where the standard linear FDT is replaced by a nonlinear, bilinear-response counterpart relating commutator correlators to response functions of products of observables—allowing experimental access to long-range NESS correlations and generalized rigidity (Kirkpatrick et al., 2023).

Despite these advances, significant challenges remain in classifying FDT violations in strongly nonergodic, memory-dominated, or active matter systems, and in constructing operator-based quantum FDTs beyond linear response (Gomes-Filho et al., 2023). The fluctuation–dissipation paradigm continues to evolve, with emerging connections to quantum thermodynamics, metrological limits (e.g., via quantum Fisher information and SLD), and hydrodynamics in strongly correlated and driven systems (Mehboudi et al., 2017, Safi, 12 Oct 2025).

Table: Representative Fluctuation–Dissipation Theorem Regimes and Generalizations

Regime/Class	FDT Formulation	Notable Features/References
Classical equilibrium	$S(\omega) = 2k_B T\, \text{Im} \chi(\omega)/\omega$	Langevin, Onsager, Kubo; Einstein relation (Gomes-Filho et al., 2023)
Quantum equilibrium	$S(\omega) = \hbar\,\chi''(\omega)\,\coth(\hbar\omega/2k_B T)$	Kubo–Martin–Schwinger, BE factor (Branchina et al., 2010)
NESS (classical/quantum)	$R = R_{\text{relax}} + R_{\text{housekeeping}}$	Curl flux, gauge field, entropy production (Feng et al., 2011, Zhang et al., 2021)
Lévy/Non-Gaussian	$\chi(t) = d/dt \langle X(t)X(0) \rangle$	Nonlinear conjugate, infinite variance (Dybiec et al., 2012)
Quantum Violation	$\| \sigma_{xy}(0) - \beta S_{xy}(0) \| \gg \| \sigma_{xy}(0) \|$	Macroscopic violation; antisymmetric part (Kubo et al., 2018)
Growth phenomena	$D = b\nu w_s^2$ (hidden FDT)	KPZ, EW interfaces, universal roughness (Gomes-Filho et al., 2021)
Machine learning SGD	$\Sigma_{xx}\Sigma_{WW}+\Sigma_{WW}\Sigma_{xx}=2D$	Parameter variance in linear regression (Han et al., 2021)

The FDT thus encapsulates a universal bridge between noise and response—its precise form and limitations reflecting the interplay of detailed balance, ergodicity, quantum effects, and non-equilibrium driving across disciplines from condensed matter to machine learning.