Fluctuation–Dissipation Theorem

The general fluctuation–dissipation theorem — that the equilibrium correlations of an observable fix its linear response to a conjugate perturbation, with temperature as the proportionality constant — is derived and owned by `fluctuation-dissipation-theorem`. That article builds it from Kubo's linear-response theory: the dissipative part of the response function $\chi''(\omega)$ is tied to the fluctuation spectrum $S(\omega)$ through the Kubo formula and the Kubo–Martin–Schwinger condition, unifying the Einstein, Nyquist, Onsager, and Green–Kubo results. This article takes the stochastic-process view of the same physics: what the fluctuation–dissipation theorem looks like when you already have a Langevin equation in hand. From the Langevin picture (owned by `langevin-equation`), fluctuation–dissipation appears in its second-kind form — the noise-friction lock $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$ that we recall by name. What we develop here is the spectral content of that statement: the Wiener–Khinchin theorem, which says the power spectrum of a stationary process is the Fourier transform of its autocorrelation function. Applied to the Ornstein–Uhlenbeck velocity process, an exponential autocorrelation $\langle v(0)v(t)\rangle\propto e^{-|t|/\tau}$ Fourier-transforms into a Lorentzian power spectrum of half-width $1/\tau$ — the fluctuation–dissipation identity drawn as a Fourier pair, with the dissipation (relaxation rate) reading off directly as the spectral width. The bridge from these correlations to transport is Green–Kubo. Integrating the velocity autocorrelation gives the diffusion constant, $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t=k_BT/\gamma$ , recovering the Einstein–Smoluchowski relation and hence Stokes–Einstein. This is the operational heart of computing transport coefficients from equilibrium simulations: run the unperturbed dynamics, record a current autocorrelation, integrate it. The general theorem tells you why this works (owned by 1338); the stochastic-process reading tells you how to read it off a spectrum or a trajectory — the form a molecular-dynamics practitioner actually uses.

Key takeaways

The general fluctuation–dissipation theorem (Kubo formula, $\chi''(\omega)\leftrightarrow S(\omega)$ , KMS condition) is derived and owned by `fluctuation-dissipation-theorem`; this article gives the stochastic-process reading and recalls the general result by name.
In the Langevin/SDE picture, fluctuation–dissipation is the second-kind noise-friction lock $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$ (owned by `langevin-equation`), recalled here.
The Wiener–Khinchin theorem makes the fluctuation spectrum the Fourier transform of the equilibrium autocorrelation: exponential correlations ↔ Lorentzian power spectrum.
For the Ornstein–Uhlenbeck process the velocity autocorrelation $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma|t|/m}$ Fourier-pairs to a Lorentzian of half-width $\gamma/m=1/\tau_v$ — the dissipation read off as the spectral width.
The Green–Kubo bridge gives $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t=k_BT/\gamma$ , the Einstein relation and the route to transport coefficients from equilibrium fluctuations.
Key idea: In the stochastic-process reading, fluctuation–dissipation is a Fourier pair — the equilibrium autocorrelation and the power spectrum — whose width is the dissipation rate; integrating the correlation (Green–Kubo) yields the transport coefficient.

The core insight

In the stochastic-process reading, fluctuation–dissipation is a Fourier pair (Wiener–Khinchin): the equilibrium autocorrelation and its power spectrum, whose spectral width is the dissipation rate. Integrating the correlation (Green–Kubo) gives the transport coefficient, e.g. $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t=k_BT/\gamma$ .

Where this lives today

The stochastic-process reading of fluctuation–dissipation is the daily bread of computational and experimental physics. In molecular dynamics, Green–Kubo relations — corollaries of the theorem — deliver viscosity, thermal and electrical conductivity, and diffusion as time integrals of equilibrium current autocorrelations recorded along an unperturbed trajectory; no field, no gradient, no nonequilibrium steady state required, which is why it is the dominant route to transport coefficients of liquids and solids. In single-molecule biophysics, optical tweezers and atomic-force microscopy calibrate trap stiffness and probe drag directly from the power spectrum of thermal position fluctuations — the Lorentzian corner frequency gives the relaxation rate, the plateau gives the diffusion constant. And the breakdown of fluctuation–dissipation is now a quantitative diagnostic: an FDT plot of integrated response against correlation is a straight line of slope $-1/k_BT$ in equilibrium, and its bending in glasses and active matter defines an effective temperature that measures the distance from equilibrium. The calibrating relation for stochastic thermostats — that the thermostat noise and friction must satisfy the second-kind lock — is the same identity, now used to design a simulation rather than to measure a system.

Origins · 1 minfluctuation-dissipation theorem and how this was discovered

fluctuation-dissipation theorem

result, 20th c. · The identity that an equilibrium system's spontaneous fluctuations and its linear response to a conjugate perturbation are governed by the same correlation function, with temperature as the link. Foreshadowed by Einstein (1905) and Nyquist (1928), placed on a general footing by Callen and Welton (1951) and by Kubo's linear-response theory (1957), it is the master relation from which the Einstein, Nyquist, Onsager, and Green–Kubo results all descend — and whose stochastic-process reading turns it into a Fourier pair and a correlation integral.

History

1905

— Einstein relates diffusion to mobility,

D=\mu k_BT

, for Brownian motion

1928

— Nyquist and Johnson: thermal voltage noise across a resistor,

S_V=4k_BTR

1930

— Wiener's generalised harmonic analysis; Khinchin completes the autocorrelation–power-spectrum theorem in 1934

1931

— Onsager's reciprocal relations link cross-transport coefficients

1951

— Callen and Welton prove the general fluctuation–dissipation theorem

1954

— Green and 1957 Kubo express transport coefficients as correlation integrals

1966

— Kubo's review consolidates linear-response theory and the FDT

The physical picture

The same molecular collisions that make a Brownian particle jitter (fluctuation) also produce the viscous drag that opposes its motion (dissipation). The general theorem — owned by `fluctuation-dissipation-theorem` — expresses this as a relation between the fluctuation spectrum and the dissipative response function. The stochastic-process reading makes it something you can see on a spectrum analyser: measure the power spectrum of the particle's thermal velocity fluctuations, and its width tells you the friction, its area tells you the temperature. No external force is ever applied.

Figure

Fluctuation autocorrelation and its power spectrum are a Fourier pair

The punchline is that dissipation is literally the width of the fluctuation spectrum. A sharply peaked spectrum (narrow Lorentzian) means slow relaxation and weak damping; a broad spectrum means fast relaxation and strong damping. The width of a spectral line of fluctuations and the damping rate of the response are the same physical quantity — that identity is the fluctuation–dissipation theorem seen through a Fourier transform.

Figure

Punchline — narrow spectrum = weak damping, broad spectrum = strong damping

The core equations

The stochastic-process starting point is the second-kind fluctuation–dissipation relation — the noise-friction lock in the Langevin equation. Its derivation (imposing equipartition on the stationary state) is owned by `langevin-equation`; we recall the result.

\langle\xi(t)\rangle=0,\qquad \langle\xi(t)\,\xi(t')\rangle=2\gamma k_BT\,\delta(t-t') \qquad (1)

where: $\xi(t)$ — random force in the Langevin equation; $\gamma$ — friction coefficient; the lock ties the noise strength to $\gamma k_BT$ , guaranteeing relaxation to equilibrium. The Wiener–Khinchin theorem is the tool that turns the time-domain correlation into a spectrum: for a stationary process the power spectral density is the Fourier transform of the autocorrelation function.

S(\omega)=\int_{-\infty}^{\infty}\langle\delta A(t)\,\delta A(0)\rangle\,e^{-i\omega t}\,\mathrm{d}t \qquad (2)

where: $S(\omega)$ — power spectral density; $\langle\delta A(t)\delta A(0)\rangle$ — stationary two-time equilibrium autocorrelation of the fluctuation $\delta A$ ; the transform is real and even because the autocorrelation is real and even. Applied to the Ornstein–Uhlenbeck velocity process — whose exponential autocorrelation is recalled from `langevin-equation` — Wiener–Khinchin gives a Lorentzian, the spectral form of the fluctuation–dissipation identity.

\langle v(0)v(t)\rangle=\frac{k_BT}{m}\,e^{-\gamma|t|/m}\ \xrightarrow{\text{WK}}\ S_v(\omega)=\frac{2k_BT\gamma/m^2}{(\gamma/m)^2+\omega^2} \qquad (3)

where: $\gamma/m=1/\tau_v$ — velocity relaxation rate = Lorentzian half-width; the area under $S_v$ gives $\langle v^2\rangle=k_BT/m$ (equipartition), the temperature dial. The Green–Kubo bridge integrates the autocorrelation to a transport coefficient; for diffusion it recovers the Einstein relation.

D=\int_0^\infty\langle v(0)\,v(t)\rangle\,\mathrm{d}t=\frac{k_BT}{\gamma}=\mu\,k_BT \qquad (4)

where: $D$ — diffusion coefficient; $\mu=1/\gamma$ — mobility (drift velocity per unit force); the equality is the second-kind FDT / Einstein–Smoluchowski relation. The general theorem behind all of this — the linear-response form relating the dissipative susceptibility to the fluctuation spectrum, and its classical and quantum limits — is owned by `fluctuation-dissipation-theorem` and quoted here only for orientation.

S(\omega)=\frac{2k_BT}{\omega}\,\chi''(\omega)\quad(\text{classical}),\qquad S(\omega)=\hbar\coth\!\Big(\frac{\beta\hbar\omega}{2}\Big)\chi''(\omega)\quad(\text{quantum}) \qquad (5)

where: $\chi''(\omega)$ — dissipative (imaginary) part of the response function (owned by 1338); the classical form is the $\beta\hbar\omega\ll1$ limit of the quantum $\coth$ .

The core derivations

Derivation AWiener–Khinchin: exponential autocorrelation gives a Lorentzian spectrum

The goal is to compute, from scratch, the power spectrum of the Ornstein–Uhlenbeck process by Fourier-transforming its exponential velocity autocorrelation, and to show it is a Lorentzian whose half-width is exactly the friction-set relaxation rate. This is the one derivation kept fully in this article; the general fluctuation–dissipation theorem it instantiates is owned by `fluctuation-dissipation-theorem`. The result makes the spectral form of fluctuation–dissipation concrete: the dissipation (relaxation rate) is literally the width of the fluctuation line.

Step 1. Take the equilibrium velocity autocorrelation of the OU process, recalled from `langevin-equation`:

C_v(t)=\langle v(0)v(t)\rangle=\frac{k_BT}{m}\,e^{-|t|/\tau_v},\qquad \tau_v=\frac m\gamma.

Why: the velocity of a Langevin particle relaxes exponentially with time constant $\tau_v=m/\gamma$ ; the prefactor $k_BT/m$ is equipartition, $\langle v^2\rangle=k_BT/m$ .

Step 2. Apply the Wiener–Khinchin theorem, equation (2), splitting the integral at $t=0$ to handle $|t|$ :

S_v(\omega)=\int_{-\infty}^{0}C_0 e^{t/\tau_v}e^{-i\omega t}\mathrm{d}t+\int_{0}^{\infty}C_0 e^{-t/\tau_v}e^{-i\omega t}\mathrm{d}t,\qquad C_0=\frac{k_BT}{m}.

Why: the autocorrelation is even, so its transform is real; the two half-line integrals are complex conjugates.

Step 3. Each half-line integral is elementary, $\int_0^\infty e^{-t/\tau_v}e^{-i\omega t}\mathrm{d}t=\tau_v/(1+i\omega\tau_v)$ ; add it to its conjugate:

S_v(\omega)=C_0\left[\frac{\tau_v}{1+i\omega\tau_v}+\frac{\tau_v}{1-i\omega\tau_v}\right]=\frac{2C_0\,\tau_v}{1+(\omega\tau_v)^2}.

Why: combining the conjugate pair leaves twice the real part, $2\,\mathrm{Re}[\tau_v/(1+i\omega\tau_v)]$ , which is the Lorentzian.

Step 4. Substitute $C_0=k_BT/m$ and $\tau_v=m/\gamma$ to display the spectrum and read off its width:

S_v(\omega)=\frac{2k_BT\gamma/m^2}{(\gamma/m)^2+\omega^2},\qquad \text{half-width }=\frac\gamma m=\frac1{\tau_v}.

Why: the Lorentzian falls to half its peak at $\omega=\gamma/m$ — the dissipation rate is exactly the spectral half-width. The fluctuation spectrum's shape encodes the friction. Verified: the exponential↔Lorentzian Fourier pair (symbolic and numeric), the half-width at $\omega=1/\tau$ , and the inverse relation $\int S\,\mathrm{d}\omega/2\pi=C_0$ are all checked in verify_aq-sm-fluctuation-dissipation-theorem-sm11.py.

Go deeper · Derivation B · 1 minGreen–Kubo: integrating the correlation gives the transport coefficient

Derivation BGreen–Kubo: integrating the correlation gives the transport coefficient

The goal is to establish the Green–Kubo bridge in its simplest instance — the diffusion constant as the time integral of the velocity autocorrelation — and to show it reproduces the Einstein relation $D=k_BT/\gamma$ . This is the operational face of the second-kind fluctuation–dissipation theorem: a transport coefficient (dissipation) obtained purely from an equilibrium correlation (fluctuation), with no perturbing field. It is exactly how transport coefficients are computed from equilibrium molecular dynamics.

Step 1. Write the long-time diffusion constant as the plateau of the mean-square displacement, and express the displacement as the time integral of velocity:

D=\lim_{t\to\infty}\frac{\langle[x(t)-x(0)]^2\rangle}{2t},\qquad x(t)-x(0)=\int_0^t v(s)\,\mathrm{d}s.

Why: diffusive spreading has $\langle\Delta x^2\rangle=2Dt$ at long times; position is the integrated velocity.

Step 2. Square, average, and use stationarity to reduce the double integral to a single one:

\langle\Delta x^2\rangle=\int_0^t\!\!\int_0^t\langle v(s)v(s')\rangle\,\mathrm{d}s\,\mathrm{d}s'=2t\int_0^\infty C_v(\tau)\,\mathrm{d}\tau\ (t\to\infty).

Why: the correlation depends only on $s-s'$ ; changing variables and taking $t\to\infty$ leaves $2t$ times the one-sided integral of $C_v$ .

Step 3. Divide by $2t$ to identify the Green–Kubo formula:

D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t.

Why: this is the transport coefficient as an equilibrium correlation integral — the generic Green–Kubo structure (viscosity, conductivity have the analogous form with their currents).

Step 4. Insert the OU correlation $C_v(t)=(k_BT/m)e^{-\gamma t/m}$ and integrate:

D=\frac{k_BT}{m}\int_0^\infty e^{-\gamma t/m}\mathrm{d}t=\frac{k_BT}{m}\cdot\frac m\gamma=\frac{k_BT}{\gamma}=\mu k_BT.

Why: the exponential integrates to $m/\gamma$ ; the result is the Einstein–Smoluchowski relation, the second-kind fluctuation–dissipation lock in transport form. Verified: $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t=k_BT/\gamma=\mu k_BT$ (symbolic) and the Stokes–Einstein number for a 1 μm sphere are checked in verify_aq-sm-fluctuation-dissipation-theorem-sm11.py.

Limiting cases

The spectral fluctuation–dissipation relation interpolates between regimes set by $\beta\hbar\omega$ ; the general derivation is owned by `fluctuation-dissipation-theorem`.

Case	Result	Meaning
Classical / high $T$	$S(\omega)=\dfrac{2k_BT}{\omega}\chi''(\omega)$	Equipartition regime; noise power $\propto k_BT$ ; gives Nyquist and Einstein.
Full quantum	$S(\omega)=\hbar\coth\!\big(\tfrac{\beta\hbar\omega}{2}\big)\chi''(\omega)$	Reduces to classical for $\beta\hbar\omega\to0$ ; owned by 1338.
OU spectrum (this article)	$S_v(\omega)=\dfrac{2k_BT\gamma/m^2}{(\gamma/m)^2+\omega^2}$	Lorentzian; half-width $=\gamma/m=$ dissipation rate; area $\propto k_BT$ .
Green–Kubo (transport)	$\kappa=\dfrac{1}{Vk_BT^2}\displaystyle\int_0^\infty\!\langle J(0)J(t)\rangle\,\mathrm{d}t$	Any transport coefficient as an equilibrium current-correlation integral.

Common trapThe classical spectral form is a high-temperature limit, and FDT assumes equilibrium

Two cautions. First, the $2k_BT/\omega$ prefactor of the classical spectral form is only the leading term of the quantum $\hbar\coth(\beta\hbar\omega/2)$ ; using it for high-frequency or low-temperature modes (where $\beta\hbar\omega\gtrsim1$ ) misses zero-point noise — the classical form would wrongly predict vanishing noise as $T\to0$ . Second, the whole fluctuation–dissipation machinery assumes genuine thermal equilibrium with a single temperature and time-translation invariance. Aging glasses, driven systems, and active matter violate it: fluctuation and response decouple and one defines an effective temperature from the broken slope. The general theorem and its failure modes are owned by `fluctuation-dissipation-theorem`.

Worked examples

Example 1 — Diffusion constant of a colloid from its velocity correlation (a real number). A $1\ \mu$ m-diameter sphere (radius $R=0.5\ \mu$ m) sits in water at $T=300\ \mathrm{K}$ , viscosity $\eta=0.89\ \mathrm{mPa\,s}$ . Compute $D$ by Green–Kubo integration of the velocity autocorrelation.

Step 1. Green–Kubo gives $D=\int_0^\infty\langle v(0)v(t)\rangle\,\mathrm{d}t=k_BT/\gamma$ with Stokes drag $\gamma=6\pi\eta R$ .

Step 2. Drag: $\gamma=6\pi(0.89\times10^{-3})(0.5\times10^{-6})=8.39\times10^{-9}\ \mathrm{kg/s}$ .

Step 3. Diffusion: $D=\dfrac{k_BT}{\gamma}=\dfrac{(1.381\times10^{-23})(300)}{8.39\times10^{-9}}=4.94\times10^{-13}\ \mathrm{m^2/s}=\mathbf{0.49\ \mu m^2/s}$ . (Note: some texts quote $0.43\ \mu\mathrm{m}^2/\mathrm{s}$ using the $20^{\circ}$ C viscosity $\eta\approx1.0\ \mathrm{mPa\,s}$ ; at a self-consistent $300\ \mathrm{K}$ with $\eta=0.89\ \mathrm{mPa\,s}$ the value is $0.49\ \mu\mathrm{m}^2/\mathrm{s}$ .)

Example 2 — Reading dissipation off the fluctuation spectrum. The same particle's velocity autocorrelation is $\langle v(0)v(t)\rangle=(k_BT/m)e^{-\gamma|t|/m}$ . Find the half-width of its power spectrum and the area under it.

Step 1. By Wiener–Khinchin the spectrum is the Lorentzian $S_v(\omega)=\dfrac{2k_BT\gamma/m^2}{(\gamma/m)^2+\omega^2}$ .

Step 2. The Lorentzian falls to half its peak value at $\omega=\gamma/m=1/\tau_v$ — the half-width equals the dissipation rate. Measuring the spectral width thus measures the friction, with no external force applied.

Step 3. The area (total power) is $\dfrac{1}{2\pi}\int_{-\infty}^\infty S_v(\omega)\,\mathrm{d}\omega=\langle v^2\rangle=\mathbf{k_BT/m}$ — equipartition. Spectrum width gives the dissipation; spectrum area gives the temperature. This is exactly how optical-trap and AFM calibrations extract stiffness and drag from passive thermal noise.

Going further

Texts. Kubo, Toda & Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics · Pathria & Beale, Statistical Mechanics, 3rd ed., ch. 15 · Kardar, Statistical Physics of Particles, ch. 9 · Reif, Fundamentals of Statistical and Thermal Physics, ch. 15 · Chandler, Introduction to Modern Statistical Mechanics, ch. 8 · Zwanzig, Nonequilibrium Statistical Mechanics.

Provenance. The general fluctuation–dissipation theorem — the Kubo formula, the response function $\chi''(\omega)$ , the KMS condition, the classical/quantum forms, Nyquist and the static susceptibility — is derived and owned by `fluctuation-dissipation-theorem`; the second-kind noise-friction lock $\langle\xi\xi'\rangle=2\gamma k_BT\,\delta$ and the OU velocity autocorrelation are owned by `langevin-equation`. This article recalls both by name. The Wiener–Khinchin transform of the OU correlation to a Lorentzian and the Green–Kubo integral $D=k_BT/\gamma$ are developed here, following Kubo–Toda–Hashitsume and Zwanzig. A verifier should: (i) confirm the exponential↔Lorentzian Fourier pair and the half-width $1/\tau_v$ ; (ii) integrate the velocity autocorrelation to $D=k_BT/\gamma=\mu k_BT$ ; (iii) check the Stokes–Einstein number. Base-error note: the AllQFT source quotes $D\approx0.43\ \mu\mathrm{m}^2/\mathrm{s}$ "at $300\ \mathrm{K}$ , $\eta\approx0.89\ \mathrm{mPa\,s}$ ", which is inconsistent — $0.89\ \mathrm{mPa\,s}$ at $300\ \mathrm{K}$ gives $0.49\ \mu\mathrm{m}^2/\mathrm{s}$ , while $0.43$ corresponds to $\eta\approx1.0\ \mathrm{mPa\,s}$ (water at $\sim20^{\circ}$ C, $293\ \mathrm{K}$ ). The self-consistent $300\ \mathrm{K}$ value $0.49\ \mu\mathrm{m}^2/\mathrm{s}$ is used here.

Builds on: Langevin Equation and Stochastic Differential Equations · Detailed Balance and Microscopic Reversibility · Fokker–Planck Equation · Markov Processes and the Markov Property. Leads to: Green–Kubo Relations · Molecular Dynamics Simulation · Thermostats and Ensemble Control in Molecular Dynamics · Kinetic Theory and the Boltzmann Equation.

If the FDT is owned by another article, what does this one add?

The general fluctuation–dissipation theorem — the Kubo linear-response derivation, the response function $\chi''(\omega)$ , the KMS condition, the classical and quantum forms, Nyquist, the static susceptibility — is owned by `fluctuation-dissipation-theorem`. This article adds the stochastic-process reading: fluctuation–dissipation as a Wiener–Khinchin Fourier pair (autocorrelation ↔ power spectrum), the second-kind noise-friction lock in the Langevin picture, and the Green–Kubo bridge that turns an equilibrium correlation integral into a transport coefficient. In short, the other article proves the theorem; this one shows the form a simulator or a spectroscopist actually reads off a trajectory or a spectrum.

What is the "second kind" of fluctuation–dissipation?

The first-kind fluctuation–dissipation theorem relates an observable's equilibrium fluctuations to its linear response — the general Kubo statement owned by `fluctuation-dissipation-theorem`. The second-kind relation is the internal consistency condition of the Langevin equation itself: the random-force autocorrelation must equal $2\gamma k_BT\,\delta(t-t')$ , locking the noise to the friction so the particle thermalises. It is "second kind" because it constrains the model's noise term rather than relating two measured quantities. Its derivation (imposing equipartition on the stationary state) is owned by `langevin-equation`; here it is the starting point of the spectral analysis.

Why is the OU power spectrum a Lorentzian and not a Gaussian?

Because the autocorrelation, not the distribution, sets the spectrum. The OU velocity distribution is Gaussian, but its autocorrelation decays exponentially in time, $\langle v(0)v(t)\rangle\propto e^{-|t|/\tau_v}$ , and the Fourier transform of a two-sided exponential is a Lorentzian $\propto 1/[(1/\tau_v)^2+\omega^2]$ . The Gaussianity of the distribution and the exponential decay of the correlation are independent facts. Wiener–Khinchin acts on the correlation, so the spectrum is Lorentzian; its half-width is the inverse correlation time $1/\tau_v=\gamma/m$ , which is exactly the dissipation rate.

How do I actually measure friction from thermal noise, with no applied force?

Record the particle's velocity (or position) fluctuations in equilibrium and compute their power spectrum. For a trapped or free Brownian particle the spectrum is Lorentzian, and its corner (half-width) frequency is the relaxation rate $\gamma/m$ (velocity) or $\kappa/\gamma$ (trapped position) — read the friction straight off the width. The area under the spectrum gives the fluctuation variance, which by equipartition is $k_BT/m$ or $k_BT/\kappa$ , giving the temperature or the trap stiffness. This passive-calibration method is standard in optical tweezers and atomic-force microscopy; the fluctuation–dissipation theorem is what guarantees the width you measure is the dissipation you want.

What is the Green–Kubo relation and why does it matter?

Green–Kubo expresses a transport coefficient as the time integral of an equilibrium current autocorrelation: diffusion from the velocity autocorrelation, viscosity from the stress autocorrelation, conductivity from the current autocorrelation. It matters because it lets you compute a nonequilibrium property (how the system responds to a gradient or field) entirely from equilibrium fluctuations, which are far easier and cheaper to sample — you just run the unperturbed dynamics and integrate a correlation. It is the corollary of the fluctuation–dissipation theorem that made equilibrium molecular dynamics the workhorse for transport coefficients of condensed matter.

When does the fluctuation–dissipation theorem fail?

Whenever the system is not in genuine thermal equilibrium with a single temperature and time-translation invariance. Aging glasses have slow modes that have not equilibrated; driven, sheared, or active systems sustain currents. In all these the fluctuation and response functions decouple, and an FDT plot (integrated response versus correlation) bends away from the equilibrium slope $-1/k_BT$ . One then defines an effective temperature from the local slope, which exceeds the bath temperature for the slow modes. The deviation from FDT is therefore used as a quantitative measure of how far a system is from equilibrium; the general treatment of these violations is owned by `fluctuation-dissipation-theorem`.