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Fluctuation–dissipation theorem

The fluctuation–dissipation theorem (FDT) is a central result in statistical physics that establishes a precise relation between the spontaneous fluctuations of a physical system in thermal equilibrium and its linear response to small external perturbations, quantifying how dissipative processes arise from these equilibrium fluctuations.^[1] This theorem, often expressed through equations linking correlation functions of fluctuating quantities to response functions like susceptibility or impedance, underpins the understanding of noise and dissipation in diverse systems, from electrical circuits to quantum materials.^[1] It reveals that the same microscopic mechanisms driving random thermal motions also govern a system's tendency to return to equilibrium after disturbance, providing a bridge between equilibrium statistical mechanics and nonequilibrium dynamics.^[2] The origins of the FDT trace back to early 20th-century studies of Brownian motion, where Albert Einstein and Marian Smoluchowski in 1905–1908 derived relations connecting diffusive fluctuations to frictional dissipation in colloidal particles suspended in fluids.^[3] A pivotal formulation emerged in 1928 when Harry Nyquist applied thermodynamic principles to calculate the thermal electromotive force in electrical conductors, deriving the now-famous Nyquist noise formula that equates voltage fluctuations to resistance and temperature: \overline{e^2} = 4kT R \Delta f, where k is Boltzmann's constant, T is temperature, R is resistance, and \Delta f is bandwidth.^[4] This work marked the first explicit fluctuation-dissipation relation in electronics, emphasizing that observable noise is an inevitable consequence of thermal agitation.^[3] Subsequent generalizations expanded the theorem's scope. In 1931, Lars Onsager's reciprocity relations laid groundwork by linking transport coefficients to fluctuation symmetries in near-equilibrium systems.^[2] The quantum version was established in 1951 by Herbert Callen and Theodore Welton, who used quantum perturbation theory to relate spectral densities of fluctuations to imaginary parts of response functions, applicable to fields like quantum optics and solid-state physics. Ryogo Kubo provided a comprehensive statistical mechanical proof in the 1950s–1960s via linear response theory, showing that the response function \chi(\omega) satisfies \chi''(\omega) = \frac{1 - e^{-\beta \hbar \omega}}{2\hbar} S(\omega), where S(\omega) is the power spectrum of fluctuations and \beta = 1/[kT](/page/kT), thus unifying classical and quantum cases.^[1] The FDT's importance extends far beyond theory, enabling practical predictions of thermal noise in amplifiers and sensors, analysis of Brownian dynamics in biological processes like molecular motors, and modeling of dissipation in glassy and granular materials far from equilibrium.^[2] Violations or modifications of the FDT signal departures from equilibrium, as seen in active matter systems or driven quantum devices, and continue to drive research in complex systems, including the interplay of disorder and fluctuations recognized in Giorgio Parisi's 2021 Nobel Prize work.^[2] Today, the theorem remains a cornerstone for interpreting experimental data in fields ranging from condensed matter to biophysics, with ongoing extensions to nonlinear and open quantum systems.^[2]

Introduction

Qualitative Overview

The fluctuation-dissipation theorem establishes a profound link between the spontaneous thermal fluctuations in a physical system at equilibrium and its dissipative response to small external perturbations, demonstrating that noise and dissipation are intimately connected aspects of the same underlying thermal processes.^[5] This principle reveals that the random variations inherent to any system in contact with a heat bath directly determine how it absorbs or dissipates energy when subjected to weak influences, providing a unified view of equilibrium dynamics.^[6] Conceptually, one can think of fluctuations as incessant, random jostling from the environment that keeps the system in thermal balance, akin to microscopic "kicks" that prevent stagnation, while dissipation acts as the frictional drag or energy leakage to that same environment when an external force is applied. The theorem quantifies this duality by showing that the strength of these fluctuations is proportional to the dissipative rate, scaled by the system's temperature, underscoring their equivalence in maintaining equilibrium.^[7] In physics, this theorem serves as a cornerstone for connecting the probabilistic realm of statistical mechanics—governing the collective behavior of countless particles—to the deterministic laws of macroscopic thermodynamics, enabling predictions of fluctuation amplitudes from easily measured transport properties like mobility or conductivity.^[8] It thus facilitates practical applications, such as estimating thermal noise in resistors from their ohmic resistance or diffusion rates in Brownian motion from frictional coefficients. The theorem's scope is confined to systems in thermal equilibrium at temperature T, where the energy scale k_B T (with k_B Boltzmann's constant) dictates the overall magnitude of both fluctuations and responses.^[6]

Historical Context

The fluctuation-dissipation theorem traces its origins to early 20th-century investigations into stochastic processes in physical systems. In 1905, Albert Einstein published a seminal paper on Brownian motion, establishing a quantitative relation between the random diffusive fluctuations of suspended particles and the dissipative viscous drag exerted by the surrounding fluid, thereby providing a foundational link between microscopic thermal agitation and macroscopic transport properties. This work laid the groundwork for understanding how equilibrium fluctuations relate to dissipative phenomena, influencing subsequent developments in statistical mechanics. Two decades later, in 1928, Harry Nyquist extended these ideas to electrical systems by deriving a theorem on thermal noise in resistors, demonstrating that voltage fluctuations across a conductor are directly proportional to its resistance and temperature, thus connecting electrical dissipation to thermal equilibrium noise. Building on these precursors, Lars Onsager contributed a crucial theoretical foundation in 1931 through his reciprocity relations for irreversible processes, which incorporated fluctuations to derive symmetric transport coefficients under time-reversal symmetry, unifying aspects of dissipation and stochastic behavior in nonequilibrium thermodynamics. The formal statement of the fluctuation-dissipation theorem emerged in 1951 with the work of Herbert B. Callen and Theodore A. Welton, who generalized the relation within quantum statistical mechanics, showing that response functions to external perturbations are analytically connected to the power spectrum of equilibrium fluctuations, thereby encompassing and unifying the classical cases from Einstein and Nyquist. The theorem's evolution continued in 1957 when Ryogo Kubo developed linear response theory, providing a broader statistical-mechanical framework that extended the fluctuation-dissipation relations to a wide class of quantum and classical systems, emphasizing their role in calculating transport coefficients from equilibrium correlations. This progression from classical mechanics in the early 1900s—rooted in empirical observations of motion and noise—to quantum electrodynamics applications in the mid-20th century solidified the theorem as a cornerstone of nonequilibrium statistical physics, enabling profound insights into the interplay between fluctuations and dissipation across diverse physical domains.^[9]

Key Examples

Brownian Motion

The Brownian motion of colloidal particles in a fluid exemplifies the fluctuation-dissipation theorem, illustrating how thermal fluctuations drive random motion while dissipative forces govern the response to external influences. Consider a spherical particle of mass m suspended in a viscous fluid at temperature T. The particle is subject to random thermal kicks from collisions with surrounding fluid molecules, modeled as a stochastic force, and a frictional drag force -\gamma \mathbf{v}, where \gamma is the friction coefficient and \mathbf{v} is the particle's velocity. This setup, analyzed by Albert Einstein in 1905, treats the motion as overdamped for small particles, where inertial effects are negligible compared to viscous drag.^[10] The fluctuation aspect manifests in the particle's diffusive behavior, quantified by the mean-squared displacement \langle x^2(t) \rangle = 2Dt in one dimension, where [D](/page/D*) is the diffusion constant and t is time. Drawing from the equipartition theorem, which assigns \frac{1}{2} k_B [T](/page/Temperature) of thermal energy per degree of freedom, Einstein derived [D](/page/D*) = \frac{k_B [T](/page/Temperature)}{\gamma}, relating the amplitude of random displacements directly to temperature and friction. This expression highlights how thermal noise—arising from molecular agitation—produces observable diffusion.^[10] The dissipation aspect concerns the particle's deterministic response to an applied force \mathbf{F}, yielding a steady-state drift velocity \mathbf{v} = \mu \mathbf{F}, where the mobility \mu = \frac{1}{\gamma} measures how easily the particle moves against friction. The fluctuation-dissipation theorem bridges these phenomena through the Einstein-Sutherland relation D = \mu k_B T, or equivalently D = \frac{k_B T}{\gamma}, which equates the strength of fluctuations (diffusion coefficient) to the dissipative response (mobility) via the shared friction parameter \gamma. This relation, independently anticipated by William Sutherland in 1905, underscores the thermodynamic consistency between random thermal perturbations and linear response.^[10]^[11] Jean Perrin's experiments from 1908 to 1913 provided empirical confirmation of this relation using microscopic observations of gamboge particles in water. By measuring diffusion constants and sedimentation under gravity, Perrin verified D = \frac{k_B T}{\gamma} to within experimental precision, yielding consistent estimates of Avogadro's number and bolstering the atomic hypothesis of matter. These results, detailed in his 1913 monograph Les Atomes, earned Perrin the 1926 Nobel Prize in Physics.^[12]

Thermal Noise in Resistors

In a resistor of resistance R maintained at absolute temperature T, thermal agitation of charge carriers gives rise to random voltage fluctuations across its terminals.^[4] These fluctuations, known as thermal noise or Johnson-Nyquist noise, arise from the random thermal motion of electrons in the conductor, producing a stochastic electromotive force even in the absence of an external bias.^[13] The magnitude of these voltage fluctuations is characterized by the mean-squared noise voltage \langle V^2 (\Delta f) \rangle within a frequency bandwidth \Delta f, expressed by the Nyquist formula:

\langle V^2 (\Delta f) \rangle = 4 k_B T R \Delta f,

where k_B is Boltzmann's constant.^[4] This noise appears white over a wide frequency range, meaning its power spectral density is approximately constant, and it is practically observable as a broadband hiss in sensitive electronic amplifiers or receivers.^[13] The dissipative aspect of the resistor is evident in its linear response to an applied voltage V, yielding a steady current I = V / R, where R quantifies the energy dissipation through Joule heating.^[4] The fluctuation-dissipation theorem relates these phenomena by equating the fluctuation spectrum to the dissipative response, with the voltage noise power spectral density given by S_V(\omega) = 4 k_B T \operatorname{Re}[Z(\omega)], where Z(\omega) is the complex impedance and \operatorname{Re}[Z(\omega)] corresponds to the resistive part.^[4] For a purely resistive element, \operatorname{Re}[Z(\omega)] = R, directly linking the equilibrium fluctuations to the real part of the impedance.^[14] Harry Nyquist derived this relation in 1928 using thermodynamic arguments, ensuring consistency with the second law by balancing energy exchange between resistors at equilibrium across all frequencies.^[4] The prediction was experimentally verified by J. B. Johnson through measurements of noise voltages in conductors, employing vacuum tube amplifiers to detect the fluctuations and thermocouples to quantify the heating, confirming the linear dependence on T and R.^[13] This application illustrates the theorem's role in predicting noise from dissipative properties in electrical circuits at thermal equilibrium.^[4]

Theoretical Framework

Classical Formulation

In classical statistical mechanics, the fluctuation-dissipation theorem establishes a fundamental connection between the dissipative response of a system to weak external perturbations and the intrinsic thermal fluctuations occurring in equilibrium. Consider a physical system described by a Hamiltonian H_0 in the canonical ensemble at temperature T, where an observable A(t) is monitored. A small time-dependent perturbation is introduced via an additional term -\int B(\mathbf{r}, t) h(\mathbf{r}, t) \, d\mathbf{r} in the Hamiltonian, with h(\mathbf{r}, t) as the external field and B(\mathbf{r}, t) as the conjugate operator (or dynamical variable in classical terms). The linear response of the average \langle A(t) \rangle to this perturbation is characterized by the response function \chi_{AB}(t - t') = \frac{\delta \langle A(t) \rangle}{\delta h(t')} \big|_{h=0}, which quantifies how the system dissipates energy under the influence of h. The theorem links this response to equilibrium fluctuations through the Kubo formula, which in the time domain takes the form

\chi_{AB}(t) = -\frac{1}{k_B T} \theta(t) \frac{d}{dt} \langle A(t) B(0) \rangle

for t > 0, where \theta(t) is the Heaviside step function, k_B is Boltzmann's constant, and \langle A(t) B(0) \rangle is the equilibrium time-correlation function in the unperturbed system. In the frequency domain, the Fourier transforms yield the fluctuation-dissipation relation

S_{AB}(\omega) = \frac{2 k_B T}{\omega} \operatorname{Im} \left[ \chi_{AB}(\omega) \right],

where S_{AB}(\omega) is the spectral density of the fluctuations, defined as the Fourier transform of \langle A(t) B(0) \rangle. This relation demonstrates that dissipation (captured by the imaginary part of the susceptibility) directly arises from thermal fluctuations, with the factor $2 k_B T / \omega emerging in the classical high-temperature limit. The validity of this formulation relies on key assumptions: the perturbation must be weak to ensure the linear response regime, the system remains in or near thermal equilibrium (satisfying detailed balance), and the dynamics are Markovian, allowing the use of the canonical ensemble for correlations. These conditions hold for systems where microscopic reversibility ensures that forward and backward transition rates balance in equilibrium. Special cases illustrate the theorem's utility; for instance, applying it to position-momentum correlations in Brownian motion yields the Einstein relation D = k_B T / \gamma, relating the diffusion coefficient D to the friction coefficient \gamma. Similarly, for electrical conductivity, the theorem reproduces Nyquist's result for thermal noise in a resistor, \langle V^2 \rangle = 4 k_B T R \Delta f, where R is resistance and \Delta f is bandwidth.

Quantum Formulation

In the quantum formulation of the fluctuation-dissipation theorem, physical observables are represented by operators \hat{A} and \hat{B} in the Heisenberg picture, where time evolution is governed by the commutator with the Hamiltonian. The linear response function \chi_{AB}(t) describes the change in the expectation value of \hat{A} due to a perturbation coupled to \hat{B}, given by

\chi_{AB}(t) = -\frac{i}{\hbar} \theta(t) \left\langle \left[ \hat{A}(t), \hat{B}(0) \right] \right\rangle,

with \theta(t) the Heaviside step function and the average taken over the equilibrium state. This expression arises from the Kubo linear response theory and captures the causal response of quantum systems to external fields. In the frequency domain, the theorem relates the power spectral density S_{AB}(\omega) of fluctuations to the imaginary part of the response function via

S_{AB}(\omega) = \hbar \omega \coth\left( \frac{\hbar \omega}{2 k_B T} \right) \operatorname{Im} \left[ \chi_{AB}(\omega) \right],

where k_B is Boltzmann's constant and T the temperature. This relation, originally derived by Callen and Welton, generalizes the classical theorem by incorporating quantum statistical mechanics. In the high-temperature limit where k_B T \gg \hbar |\omega|, the coth factor approximates to $2 k_B T / \hbar \omega, recovering the classical form S(\omega) = \frac{2 k_B T}{\omega} \operatorname{Im} \left[ \chi(\omega) \right]. The quantum version enforces detailed balance through the asymmetry in the spectral density: S_{AB}(-\omega) = e^{-\hbar \omega / k_B T} S_{AB}(\omega), reflecting the difference between absorption and emission processes in quantum systems at finite temperature. This principle ensures consistency with the second law of thermodynamics and accounts for the Bose-Einstein distribution of excitations. Even at zero temperature, fluctuations persist due to zero-point motion from the quantum vacuum, where the coth term approaches unity, yielding S_{AB}(\omega) = \hbar |\omega| \operatorname{Im} \left[ \chi_{AB}(\omega) \right] \operatorname{sgn}(\omega). The symmetric and antisymmetric parts of the correlation functions then separate vacuum contributions from thermal ones. In applications to quantum optics, this formulation describes noise in electromagnetic fields; for a blackbody cavity, applying the theorem to the field operators reproduces Planck's law for the spectral energy density, u(\omega, T) = \frac{\hbar \omega^3}{\pi^2 c^3} \frac{1}{e^{\hbar \omega / k_B T} - 1}, as a direct consequence of equilibrium fluctuations balanced by dissipation.

Derivations

Classical Derivation

The classical derivation of the fluctuation-dissipation theorem (FDT) can be obtained through the framework of linear response theory applied to stochastic dynamics described by the Langevin equation. Consider a particle of mass m subject to a harmonic potential with natural frequency \omega_0, viscous friction with damping coefficient \gamma, an external driving force F(t), and a fluctuating force \xi(t) arising from the thermal bath. The governing equation is

m \ddot{x}(t) + \gamma \dot{x}(t) + m \omega_0^2 x(t) = F(t) + \xi(t),

where the fluctuating force satisfies \langle \xi(t) \rangle = 0 and has a yet-to-be-determined correlation function \langle \xi(t) \xi(0) \rangle. This equation models the system in the classical regime, assuming Markovian dynamics and a weak coupling to the bath.^[6] To connect fluctuations to dissipation, first compute the linear response of the system to the external force F(t). In the absence of fluctuations (\xi(t) = 0), take the Fourier transform with convention \tilde{x}(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} x(t), yielding

-m \omega^2 \tilde{x}(\omega) + i \omega \gamma \tilde{x}(\omega) + m \omega_0^2 \tilde{x}(\omega) = \tilde{F}(\omega).

The susceptibility \chi(\omega), defined as \tilde{x}(\omega) = \chi(\omega) \tilde{F}(\omega), is then

\chi(\omega) = \frac{1}{m} \frac{1}{\omega_0^2 - \omega^2 + i (\gamma / m) \omega}.

The dissipative part is captured by the imaginary component (for \omega > 0),

\text{Im} [\chi(\omega)] = \frac{1}{m} \frac{ (\gamma / m) \omega }{ [ \omega_0^2 - \omega^2 ]^2 + [ (\gamma / m) \omega ]^2 } = \frac{ \gamma \omega / m^2 }{ [ \omega_0^2 - \omega^2 ]^2 + ( \gamma \omega / m )^2 }.

This Im[\chi(\omega)] quantifies energy dissipation under periodic driving, proportional to the friction \gamma.^[6] Next, compute the equilibrium fluctuations by solving the full Langevin equation with F(t) = 0. The velocity autocorrelation function \langle v(t) v(0) \rangle, where v(t) = \dot{x}(t), can be found by integrating the formal solution or using projection methods; for the underdamped case (\gamma < 2 m \omega_0), it decays exponentially with oscillations. In the overdamped limit or for the free particle case (\omega_0 = [0](/page/0)), it simplifies to \langle v(t) v(0) \rangle = (k_B T / m) e^{- (\gamma / m) |t| }, consistent with equipartition \langle v^2 \rangle = k_B T / m. The power spectral density of velocity fluctuations S_v(\omega), the Fourier transform of the autocorrelation,

S_v(\omega) = \int_{-\infty}^{\infty} dt \, e^{i \omega t} \langle v(t) v(0) \rangle,

yields for the free particle S_v(\omega) = 2 (k_B T / m) (\gamma / m) / [ \omega^2 + (\gamma / m)^2 ]. For the general harmonic case, the position fluctuation spectrum S_x(\omega) = \frac{2 k_B T \gamma }{ m^2 \left( [\omega_0^2 - \omega^2]^2 + \left( \frac{\gamma \omega}{m} \right)^2 \right) }, derived by ensuring the fluctuation-dissipation relation for the noise \langle \xi(t) \xi(0) \rangle = 2 \gamma k_B T \delta(t) reproduces the canonical equilibrium distribution via the Fokker-Planck equation associated with the Langevin dynamics.^[6] The linking step emerges by comparing the fluctuation spectrum to the dissipative response. Substituting the noise correlation into the Langevin equation and computing the steady-state spectrum via Green's function methods shows that S_x(\omega) = (2 k_B T / \omega) \text{Im} [\chi(\omega)]. This identity holds because the same friction \gamma governs both the deterministic decay in response and the noise-driven fluctuations, with temperature T ensuring consistency with the canonical ensemble. For velocity or other observables, analogous relations follow by differentiating or projecting. This derivation assumes ergodicity, allowing time averages to equal ensemble averages, and time-translation invariance in the equilibrium state, validating the delta-correlated noise and stationary correlations.^[6]

Quantum Derivation

The quantum linear response of an observable \hat{A} to a perturbation coupled to \hat{B} is described by the Kubo formula, which in the time domain reads \chi_{AB}(t) = \frac{i}{\hbar} \theta(t) \langle [\hat{A}(t), \hat{B}(0)] \rangle_0, where \theta(t) is the Heaviside step function, the average is over the unperturbed equilibrium state, and \hat{A}(t) evolves in the interaction picture. The Fourier transform is then \chi_{AB}(\omega) = \frac{i}{\hbar} \int_0^\infty dt \, e^{i \omega t} \langle [\hat{A}(t), \hat{B}(0)] \rangle_0. This expression arises from first-order perturbation theory applied to the density matrix in the interaction picture. To derive the fluctuation-dissipation theorem (FDT), consider the equilibrium correlations, particularly the fluctuation spectrum defined as the Fourier transform of the correlation function. The relevant quantity is S(\omega) = \int_{-\infty}^\infty dt \, e^{i \omega t} \langle \hat{A}(t) \hat{B}(0) \rangle_0, but in the spectral decomposition using the complete set of energy eigenstates |n\rangle with energies E_n and probabilities p_n = e^{-\beta E_n}/Z (where \beta = 1/k_B T and Z is the partition function), it takes the form S(\omega) = \sum_{mn} |\langle m | \hat{A} | n \rangle|^2 (p_n - p_m) \delta(\omega - E_m + E_n). This representation captures the difference in population weights, reflecting absorption and emission processes. Note that for the response function, a similar spectral decomposition yields \operatorname{Im} \chi_{AB}(\omega) = \frac{\pi}{\hbar} \sum_{mn} |\langle m | \hat{A} | n \rangle|^2 (p_n - p_m) \delta(\omega - (E_m - E_n)/\hbar), assuming \hbar = 1 in the delta argument for brevity. The derivation proceeds by inserting the complete set of states into the correlation functions and exploiting the detailed balance condition from thermal equilibrium, p_m / p_n = e^{-\beta (E_m - E_n)}. For \omega > 0, the terms in S(\omega) involve transitions where E_m > E_n, so p_n > p_m, and substituting the balance factor gives p_n - p_m = p_n (1 - e^{-\beta \hbar \omega}). Relating this to the imaginary part of the susceptibility, the fluctuation spectrum simplifies to S(\omega) = [1 + n(\omega)] 2 \hbar \operatorname{Im} [\chi_{AB}(\omega)], where n(\omega) = 1/(e^{\beta \hbar \omega} - 1) is the Bose-Einstein occupation factor. This establishes the quantum FDT, linking fluctuations to dissipative response via quantum statistical weights. For \omega < 0, the relation holds with S(\omega) = e^{\beta \hbar \omega} S(-\omega), ensuring consistency. In the high-temperature limit where \beta \hbar \omega \ll 1, the Bose factor expands as n(\omega) \approx k_B T / (\hbar \omega), so $1 + n(\omega) \approx k_B T / (\hbar \omega), yielding S(\omega) \approx (2 k_B T / \omega) \operatorname{Im} [\chi_{AB}(\omega)]. This recovers the classical FDT, confirming the quantum form as a generalization. This derivation relies on the Lehmann spectral representation and assumes weak coupling to derive linear response, valid for equilibrium systems. More rigorous formulations for interacting or nonequilibrium cases employ path-integral methods or the Keldysh contour, but the equilibrium quantum FDT remains foundational as derived here.

Extensions and Limitations

Violations in Glassy Systems

In structural glasses, such as silica or polymers, the dynamics slow dramatically below the glass transition temperature T_g, resulting in non-ergodic behavior where the system fails to explore its full phase space within observable timescales.^[15] This aging process leads to violations of the equilibrium fluctuation-dissipation theorem (FDT), as the system's response to perturbations no longer equilibrates with thermal fluctuations.^[16] The primary mechanism of FDT violation in these systems involves an effective temperature T_\mathrm{eff} > T, where T is the bath temperature, causing fluctuations to exceed equilibrium predictions. This is parameterized by the ratio of the imaginary part of the susceptibility to the power spectral density, \chi''(\omega) / S(\omega) \neq \omega / (2 k_B T), indicating a breakdown in the linear relation between response and correlation functions.^[17] In aging regimes, T_\mathrm{eff} governs the slow structural modes while the fast, local modes remain coupled to T, reflecting a separation of timescales.^[15] Experimental evidence for these violations was first observed in spin glasses through inherent noise measurements, where magnetic susceptibility responses deviated from FDT predictions below the transition temperature, consistent with early studies on aging dynamics in the 1980s and 1990s. In colloidal glasses, such as Laponite suspensions during sol-to-gel transitions, conductivity noise and polarization measurements revealed strong FDT breakdowns, with T_\mathrm{eff} reaching values up to $3 \times 10^5 K at low frequencies, accompanied by two-step relaxation processes in correlation functions.^[18] Theoretically, mode-coupling theory (MCT) predicts FDT breakdowns near the critical dynamics of the glass transition, where nonlinear feedback in density correlations leads to a two-step relaxation and an effective decoupling of slow modes, with violations emerging below the MCT transition temperature T_c.^[15] In mean-field spin glasses, replica symmetry breaking (RSB) quantifies these violations through the function X(C), which links dynamic FDT ratios to static overlap distributions, confirming RSB's role in finite-dimensional systems via numerical simulations.^[19] The introduction of a modified FDT using T_\mathrm{eff} enables probing of hidden degrees of freedom in glassy systems, with applications in granular materials where jammed packings exhibit T_\mathrm{eff} values orders of magnitude above ambient temperature, facilitating analysis of athermal quasistatic deformations. Similar extensions apply to foams, where soft glassy rheology models use T_\mathrm{eff} to describe yield-stress behavior and flow under stress.^[16] Recent studies as of 2024 on active glasses, incorporating self-propelled particles or biological motifs, have shown that applied shear can fluidize the system in certain dynamical regimes, but FDT violations persist, often characterized by an effective temperature, highlighting activity-driven effects such as rejuvenation.^[20] In 2025, further advances demonstrated that stronger active fluctuations can suppress glass transitions while more persistent fluctuations enhance them, providing new control mechanisms over glassy dynamics.^[21]

Applications in Nonequilibrium Systems

In nonequilibrium baths, where the environment exhibits time-dependent temperatures T(t) or colored noise spectra, the fluctuation-dissipation theorem (FDT) generalizes to integral relations that connect transient response functions to effective dissipation. For instance, in systems described by generalized Langevin equations with nonstationary friction kernels, the second FDT takes the form \int_0^t \chi(t-s) \, ds = \frac{k_B [T(t)](/page/Temperature)}{\gamma} for the integrated mobility \chi, linking cumulative fluctuations to instantaneous temperature variations during relaxation processes.^[22] This extension accommodates transient nonequilibrium states, such as those arising from sudden quenches or slowly varying external fields, where white noise assumptions fail.^[23] Driven systems, maintained in steady nonequilibrium states by continuous energy input like shear flows or optical pumping, exhibit violations of the standard FDT, but fluctuation relations provide analogous constraints on work and heat dissipation. In sheared fluids, for example, the steady-state power spectrum of velocity fluctuations deviates from the equilibrium form, with excess noise reflecting irreversible work input, as quantified by relations like the Jarzynski equality \langle e^{-\beta W} \rangle = e^{-\beta \Delta F}, which extends FDT principles to average exponential work fluctuations over nonequilibrium protocols.^[24] Similarly, in lasers under bias, photon number fluctuations violate the classical FDT, but generalized fluctuation theorems relate them to entropy production rates, enabling predictions of lasing thresholds from noise measurements.^[25] Active matter systems, such as bacterial suspensions or flocking models, display effective FDT-like behaviors through activity-induced effective temperatures T_\text{eff}, where self-propelled particles generate athermal fluctuations analogous to thermal ones. In dense bacterial baths, the power spectral density S(\omega) of tracer displacements satisfies S(\omega) = \frac{2 T_\text{eff}}{\omega} \operatorname{Im}[\chi(\omega)], with T_\text{eff} exceeding the ambient temperature due to motility-induced correlations, allowing rheological properties to be inferred from passive fluctuation observations.^[26] This effective framework captures collective phenomena like phase separation, where swim pressure—a nonequilibrium stress from persistent motion—mimics osmotic pressure in equilibrium gases.^[27] Theoretical generalizations of the FDT for nonequilibrium conditions include the Harada-Sasa relation, which quantifies violations in near-equilibrium driven systems by relating excess dissipation to the difference between correlation and response functions, \Delta C(t) = \frac{1}{\beta} \int_0^t e^{\beta \dot{Q}(s)} \, ds, where \dot{Q} is the heat dissipation rate.^[28] For quantum nonequilibrium noise, the Keldysh formalism provides a contour-ordered Green's function approach, deriving generalized FDTs that connect nonequilibrium spectral densities to self-energies in open quantum systems under bias or pumping.^[29] Experimental validations in the 2010s using optical tweezers on active colloids demonstrated swim pressure through force measurements on trapped particles in bacterial baths, revealing athermal contributions to confinement forces equivalent to an effective temperature scaling with activity levels.^[27] More recently, electron counting experiments in mesoscopic conductors under voltage bias, up to 2024, have tested fluctuation bounds, showing shot noise exceeding equilibrium predictions while satisfying inequality constraints like S_I \geq 2 e |I| coth forms generalized from FDT, highlighting local linear-response validity even in far-driven regimes.^[30] Recent theoretical developments as of September 2025 have generalized fluctuation-dissipation relations to active field theories, providing identities that quantify deviations from equilibrium in systems with broken time-reversal symmetry, such as nonreciprocal Cahn-Hilliard models, enabling better modeling of nonequilibrium steady states in active matter.^[31] ^[32] While no universal FDT holds in far-from-equilibrium states, local relations persist in the linear response regime around steady states, enabling effective thermodynamic descriptions for weakly driven or dilute active systems.^[33]

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These results point to a spin glass phase with spontaneously broken replica symmetry in finite dimensional spin glasses. Page 2. 1 Introduction. One of the ...
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A perspective on active glassy dynamics in biological systems
Jun 4, 2024 · Many experiments in the last two decades have revealed glassy dynamics in diverse biological systems, showing a transition between a solid-like and a fluid- ...
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On the Second Fluctuation–Dissipation Theorem for Nonequilibrium ...
Jan 1, 2014 · The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation–dissipation theorem.
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On the fluctuation-dissipation relation in non-equilibrium and non ...
Aug 30, 2019 · We review generalized fluctuation-dissipation relations, which are valid under general conditions even in “nonstandard systems,” eg, out of equilibrium and/or ...
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Nonequilibrium Equality for Free Energy Differences | Phys. Rev. Lett.
Apr 7, 1997 · Nonequilibrium Equality for Free Energy Differences. C. Jarzynski. Institute for Nuclear Theory, University of Washington, Seattle, Washington ...Missing: original | Show results with:original
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Fluctuation–dissipation relations far from equilibrium: a case study
Jun 7, 2021 · Fluctuation–dissipation relations or “theorems” (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium ...
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Fluctuations and Rheology in Active Bacterial Suspensions
Oct 3, 2007 · These measurements demonstrate violation of the fluctuation-dissipation theorem and enable us to extract the power spectrum of the active stress ...Missing: matter | Show results with:matter
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Swim Pressure: Stress Generation in Active Matter | Phys. Rev. Lett.
Jul 11, 2014 · Here we give the micromechanical basis for the swim stress and use this new perspective to study self-assembly and phase separation in active ...Missing: original | Show results with:original
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Equality Connecting Energy Dissipation with a Violation of the ...
Equality Connecting Energy Dissipation with a Violation of the Fluctuation-Response Relation. Takahiro Harada1,* and Shin-ichi Sasa2,†. 1Department of Physics ...Abstract · Article Text
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Fluctuation-dissipation theorem for non-equilibrium quantum systems
May 24, 2018 · It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain ...
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Out-of-Equilibrium Fluctuation-Dissipation Bounds | Phys. Rev. Lett.
May 3, 2024 · We prove a general inequality between the charge current and its fluctuations valid for any weakly interacting coherent electronic conductor.
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Generalized Fluctuation-Dissipation Theorem for Non-equilibrium ...
Oct 15, 2020 · The fluctuation-dissipation theorem (FDT) connecting the response of the system to external perturbations with the fluctuations at ...Abstract · Introduction · Ornstein-Uhlenbeck Process... · Conclusion