The Fluctuation-Dissipation Theorem

A resistor left alone develops a tiny hiss of thermal voltage across its terminals. A grain of pollen in water jitters without anyone touching it. Both facts are inseparable from a second fact: the resistor dissipates when a current is driven through it, and the pollen grain feels drag when pushed. The fluctuation-dissipation theorem is the exact statement that these are two faces of one molecular mechanism. The same random collisions that make an observable jiggle on its own are the collisions that drain energy when the observable is gently forced — and equilibrium fixes their ratio through a single number, the temperature. A system that fluctuates must dissipate, and by exactly the amount its fluctuations dictate. Formally the theorem lives inside linear response theory. Perturb the Hamiltonian by $H'=-B\,f(t)$ with $f$ a weak field conjugate to $B$ ; the induced response is a retarded susceptibility $\chi(t)$ given by the Kubo formula. Independently, equilibrium statistical mechanics fixes the correlation function $C(t)=\langle B(t)B(0)\rangle$ and its power spectrum $S(\omega)$ . The fluctuation-dissipation theorem is the exact algebraic identity relating the dissipative part $\chi''(\omega)$ to the fluctuation spectrum $S(\omega)$ , derived from nothing more than the canonical density matrix and time-translation invariance. Classically it reads $S(\omega)=(2k_BT/\omega)\,\chi''(\omega)$ ; quantum mechanically a factor $\coth(\hbar\omega/2k_BT)$ appears, reproducing zero-point noise as $T\to0$ . Because the derivation uses only equilibrium and linearity, the theorem is both extraordinarily robust and a sharp diagnostic. Its special cases are a roll-call of physics: Einstein's diffusion–mobility relation, the Nyquist–Johnson thermal-noise formula, the Green–Kubo transport integrals, the Langevin noise–friction balance. And whenever a measured response and a measured fluctuation spectrum fail to satisfy it, the system is not in equilibrium — a fact now exploited to define effective temperatures in glasses, to characterize molecular motors and active matter, and to test the thermostats in simulations. The theorem underpins equilibrium transport theory and, by its violation, demarcates the very boundary of equilibrium.

Key takeaways

The fluctuation-dissipation theorem (FDT) states that a system's near-equilibrium response to a weak perturbation is completely fixed by its spontaneous equilibrium fluctuations of the same observable — dissipation and fluctuation share one molecular origin.
Quantitatively, the dissipative part of the response $\chi''(\omega)$ is proportional to the fluctuation power spectrum $S(\omega)$ , with temperature $T$ the bridging constant.
Classically $S(\omega)=\dfrac{2k_BT}{\omega}\chi''(\omega)$ ; the full quantum (Callen–Welton) form carries $\coth(\hbar\omega/2k_BT)$ , which reduces to the classical result and leaves zero-point noise as $T\to0$ .
The quantum form follows from KMS detailed balance $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ , the exact equilibrium signature.
Famous special cases: the Einstein relation $D=\mu k_BT$ , the Nyquist–Johnson formula $\langle V^2\rangle=4k_BTR\,\Delta f$ , the Green–Kubo transport formulas, and the Langevin second-FDT $\langle\xi\xi'\rangle=2\gamma k_BT\,\delta$ .
Key idea: equilibrium fluctuations and near-equilibrium dissipation are governed by the same correlation function; the canonical density matrix forces their ratio to be a universal function of $\omega$ and $T$ — classically $S(\omega)=(2k_BT/\omega)\chi''(\omega)$ .

Where this lives today

The fluctuation-dissipation theorem is a working tool in three distinct communities. In simulation, the Green–Kubo formulas — direct FDT corollaries — let molecular dynamics compute diffusion constants, shear viscosities, and thermal conductivities from equilibrium-trajectory autocorrelations, avoiding the finite-size and nonlinearity artifacts of explicitly driven runs. In precision metrology, the Nyquist relation and its mechanical analogue set the fundamental noise floor of amplifiers, atomic-force microscopes, and gravitational-wave detectors: LIGO's mirror-coating thermal noise is a literal FDT calculation, and the only ways to beat it are to cool the optic or engineer a lower-loss ( $\chi''$ ) material. The most active frontier is the theorem's violation. In aging glasses, sheared suspensions, and active matter — bacterial baths, molecular motors, driven colloids — experimenters measure response $\chi$ and fluctuation spectrum $S$ independently and read off an effective temperature $T_{\rm eff}(\omega)=\omega S(\omega)/2\chi''(\omega)$ . Its deviation from the bath temperature quantifies how far the system is from equilibrium and how much energy activity injects. The FDT thus does double duty: it underpins equilibrium transport theory, and by its breakdown it draws the boundary of equilibrium itself.

The physical picture

Picture a Brownian particle. The random molecular kicks make it jiggle (fluctuation); the same collisions resist steady motion (drag). Kicks and drag come from one stream of collisions, so the strength of the jiggle and the strength of the drag cannot be set independently — equilibrium ties them with the knot $k_BT$ . Made quantitative, that single sentence is the entire theorem. The noise amplitude in the Langevin equation and its friction coefficient are not two parameters but one, welded by temperature. Think frequency by frequency. A weak oscillating force at frequency $\omega$ pumps energy into the system at a rate set by $\chi''(\omega)$ , the absorptive response. In equilibrium that absorbed energy must, on average, come back out as fluctuations at the same frequency. The bookkeeping "energy in equals energy out" pins the fluctuation spectrum to $\propto T\,\chi''(\omega)/\omega$ — a detailed balance in the frequency domain.

Figure

The classical and quantum fluctuation spectra vs frequency: as T → 0 the quantum curve does not vanish but flattens onto a zero-point plateau 2ℏχ″ — the outlier the classical theory cannot see

Callen and Welton's quantum insight, from 1951, is that even at $T=0$ a system fluctuates because of zero-point motion, and these fluctuations still control the $T=0$ dissipation. The classical $k_BT$ is replaced by $\tfrac{\hbar\omega}{2}\coth(\hbar\omega/2k_BT)$ — the mean energy of an oscillator including its ground-state $\tfrac12\hbar\omega$ . Absorption and emission become asymmetric, and their ratio is exactly the Boltzmann factor $e^{-\beta\hbar\omega}$ . Onsager's regression hypothesis is the theorem's time-domain face: a spontaneous fluctuation decays, on average, by the same macroscopic law that governs the relaxation of a deliberately prepared deviation. You cannot tell from the decay whether the bump arose by chance or by your hand — the link made quantitative in response and relaxation functions.

Figure

Nyquist noise: two very different resistors at the same T produce voltage-noise spectral densities in the exact ratio of their resistances — dissipation and noise locked together

The core equations

The starting point is the linear-response relation: the induced deviation of an observable is a causal convolution of the field with a retarded response function. This is the object the Kubo formula supplies from the microscopic dynamics.

\langle\delta A(t)\rangle=\int_{-\infty}^{t}\chi_{AB}(t-t')\,f(t')\,\mathrm{d}t' \qquad (1)

where: $\langle\delta A(t)\rangle$ — induced deviation of observable $A$ from equilibrium; $\chi_{AB}(\tau)$ — retarded response function, $=0$ for $\tau<0$ (causality); $f(t)$ — weak external field conjugate to $B$ via $H'=-B\,f(t)$ . The classical response is $\chi(\tau)=-\beta\Theta(\tau)\,\dot C(\tau)$ , tying the response to the slope of the correlation. The fluctuation side is the equilibrium correlation function and its power spectrum, the Wiener–Khinchin transform. These are properties of the undriven system; no field appears.

C_{AB}(t)=\langle A(t)\,B(0)\rangle_0,\qquad S_{AB}(\omega)=\int_{-\infty}^{\infty}C_{AB}(t)\,e^{i\omega t}\,\mathrm{d}t \qquad (2)

where: $C_{AB}(t)$ — equilibrium two-time correlation function; $S_{AB}(\omega)$ — power spectrum (Fourier transform of $C$ ). Stationarity makes $C$ even in $t$ , so $S$ is real and even — a manifestly measurable spectral density. The quantum fluctuation-dissipation theorem, in its Callen–Welton form, ties the symmetrized spectrum to the dissipative response through a $\coth$ factor. This is the master relation from which every special case falls out.

S^{\rm sym}(\omega)=\hbar\coth\!\left(\frac{\hbar\omega}{2k_BT}\right)\chi''(\omega) \qquad (3)

where: $S^{\rm sym}$ — symmetrized spectrum $\tfrac12\langle\{A(t),A(0)\}\rangle$ ; $\chi''(\omega)$ — dissipative (imaginary, absorptive) part of $\chi(\omega)$ ; the $\coth$ carries the Bose occupation $n(\omega)=1/(e^{\hbar\omega/k_BT}-1)$ plus the ground-state $\tfrac12$ . Reading it: the fluctuation spectrum is the dissipative response weighted by the mean oscillator energy per unit $\hbar\omega$ . In the classical (high-temperature) limit $\coth x\to1/x$ , and the theorem collapses to the celebrated one-line relation between spectrum and dissipation, with $k_BT$ as the sole bridge.

S(\omega)\;\xrightarrow{\;\hbar\omega\ll k_BT\;}\;\frac{2k_BT}{\omega}\,\chi''(\omega) \qquad (4)

where: $k_BT$ — thermal energy, the bridging constant. Equivalently $\chi''(\omega)=(\omega/2k_BT)S(\omega)$ : measure the fluctuation spectrum and you have measured the dissipation, and vice versa. Applied to a resistor, whose dissipative element is $R$ , the theorem is the Nyquist–Johnson formula for thermal voltage noise — the experimental cornerstone of the whole subject.

S_V(\omega)=2k_BT\,R\;\Longrightarrow\;\langle V^2\rangle=4k_BT\,R\,\Delta f \qquad (5)

where: $R$ — resistance (the dissipative element); $\Delta f$ — measurement bandwidth in Hz; $\langle V^2\rangle$ — mean-square thermal voltage. The factor $4$ (versus the two-sided $2$ ) comes from folding the spectrum into a one-sided positive-frequency band. Applied to a Langevin particle, the theorem fixes the strength of the random force in terms of the friction — the "second fluctuation-dissipation theorem." A system cannot thermalize unless its noise and drag obey this.

\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t') \qquad (6)

where: $\xi(t)$ — random (white-noise) force; $\gamma$ — friction coefficient (the dissipation). Requiring the Langevin equation to relax to Maxwell–Boltzmann ( $\tfrac12 m\langle v^2\rangle=\tfrac12 k_BT$ ) forces this amplitude — noise and friction are one parameter.

Derivation AThe classical fluctuation-dissipation theorem from canonical equilibrium

The goal is to derive $S(\omega)=(2k_BT/\omega)\chi''(\omega)$ directly from the classical Kubo response and the equilibrium correlation, using only classical statistical mechanics and integration by parts. We start from the Poisson-bracket form of the response, Fourier transform it, integrate by parts to expose the boundary term and the correlation transform, and read off the imaginary part. The strategy exploits two facts: the response is the time-derivative of the correlation (Onsager regression), and stationarity makes the correlation even so its spectrum is a cosine transform.

Step 1. Write the classical response $\chi(\tau)=-\beta\,\Theta(\tau)\,\dot C(\tau)$ with $C(\tau)=\langle A(\tau)A(0)\rangle$ and $\beta=1/k_BT$ . Why: this is the classical Kubo formula; stationarity makes $C(\tau)$ even in $\tau$ , so $\dot C(\tau)$ is odd — a fact that survives into the frequency domain.

Step 2. Fourier transform, $\chi(\omega)=-\beta\int_0^\infty\dot C(\tau)e^{i\omega\tau}\mathrm{d}\tau$ , the lower limit $0$ coming from the step function. Why: causality restricts the transform to $\tau>0$ , which is exactly what makes $\chi(\omega)$ analytic in the upper half-plane and splits it into reactive and dissipative parts.

Step 3. Integrate by parts: $\chi(\omega)=\beta\big[C(0)+i\omega\int_0^\infty C(\tau)e^{i\omega\tau}\mathrm{d}\tau\big]$ . Why: the boundary term at $\tau=0$ gives the static (real) piece $\beta C(0)$ , and the remaining integral is the half-line transform of $C$ itself — moving the derivative off $C$ and onto the exponential.

Step 4. Extract the imaginary part using $C$ even: $\chi''(\omega)=\beta\omega\int_0^\infty C(\tau)\cos(\omega\tau)\mathrm{d}\tau=\tfrac{\beta\omega}{2}S(\omega)$ , since the full-line spectrum is twice the cosine half-line transform, $S(\omega)=2\int_0^\infty C(\tau)\cos\omega\tau\,\mathrm{d}\tau$ . Why: only the $i\omega\int C e^{i\omega\tau}$ term contributes to $\mathrm{Im}\,\chi$ , and evenness of $C$ turns the transform into a cosine integral equal to $S/2$ .

Step 5. Invert: $S(\omega)=\dfrac{2}{\beta\omega}\chi''(\omega)=\dfrac{2k_BT}{\omega}\chi''(\omega)$ . Why: this is the classical FDT; because $\chi''>0$ for $\omega>0$ (passivity), the spectrum is manifestly positive, and the $\omega\to0$ limit reproduces the static sum rule $\chi(0)=\beta\langle\delta A^2\rangle$ . *Verified: verify_aq-sm-fluctuation-dissipation-theorem.py confirms the by-parts identity $\chi''=\tfrac{\beta\omega}{2}S$ on the exponential-correlation (OU) model, the static sum rule $\chi(0)=\beta\,\mathrm{Var}(A)$ , and positivity of $S(\omega)$ .*

Go deeper · Derivation B · 1 minThe quantum Callen-Welton form from KMS detailed balance

Derivation BThe quantum Callen-Welton form from KMS detailed balance

The goal is to obtain the $\coth(\hbar\omega/2k_BT)$ factor of the quantum theorem from the spectral (Lehmann) decomposition of the equilibrium correlation and response, via the detailed-balance (KMS) relation between positive- and negative-frequency spectral weight. We insert energy eigenstates into the spectrum, derive the KMS relation by relabeling dummy indices, express the dissipation as the antisymmetric spectral combination coming from the Kubo commutator, and assemble the symmetrized spectrum. The one identity that closes it is $(1+e^{-x})/(1-e^{-x})=\coth(x/2)$ .

Step 1. Write the unsymmetrized spectrum in the energy eigenbasis, $S(\omega)=\tfrac{2\pi}{Z}\sum_{m,n}e^{-\beta E_m}|A_{mn}|^2\,\delta(\omega-\omega_{nm})$ with $\omega_{nm}=(E_n-E_m)/\hbar$ . Why: this is manifestly positive (a sum of squared matrix elements weighted by Boltzmann factors) — the true, physical, non-symmetrized fluctuation spectrum.

Step 2. Derive KMS detailed balance $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ by swapping the dummy indices $m\leftrightarrow n$ : the swap sends $\omega\to-\omega$ and multiplies the Boltzmann weight by $e^{-\beta\hbar\omega}$ on the delta-support $E_n-E_m=\hbar\omega$ . Why: this exact equilibrium relation is the quantum content of detailed balance — emission ( $\omega>0$ ) outweighs absorption ( $\omega<0$ ) by the Boltzmann factor.

Step 3. Express the dissipation from the Kubo commutator as the antisymmetric spectral combination, $\chi''(\omega)=\tfrac{1}{2\hbar}[S(\omega)-S(-\omega)]$ . Why: the response function is built from $\langle[A(\tau),A(0)]\rangle$ , whose Fourier transform is the difference of the two orderings — precisely $S(\omega)-S(-\omega)$ .

Step 4. Substitute KMS: $\chi''(\omega)=\tfrac{1}{2\hbar}S(\omega)(1-e^{-\beta\hbar\omega})$ . Why: this is already the exact quantum FDT in unsymmetrized form; the factor $(1-e^{-\beta\hbar\omega})$ is what distinguishes quantum from classical.

Step 5. Symmetrize $S^{\rm sym}=\tfrac12[S(\omega)+S(-\omega)]=\tfrac12 S(\omega)(1+e^{-\beta\hbar\omega})$ and divide by $\chi''$ : using $(1+e^{-x})/(1-e^{-x})=\coth(x/2)$ gives $S^{\rm sym}(\omega)=\hbar\coth(\hbar\omega/2k_BT)\,\chi''(\omega)$ . Why: as $\beta\hbar\omega\to0$ this reduces to the classical $2k_BT/\omega$ ; as $T\to0$ , $\coth\to1$ , leaving the zero-point floor $\hbar\chi''$ — dissipation persists but no net absorption. *Verified: verify_aq-sm-fluctuation-dissipation-theorem.py confirms the KMS relation $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ , the identity $(1+e^{-x})/(1-e^{-x})=\coth(x/2)$ symbolically, the classical $\beta\hbar\omega\to0$ limit, and the $T\to0$ zero-point limit.*

Limiting cases

The single master relation collapses to many famous laws in appropriate limits — each row is a landmark theorem in disguise.

Case	Result	Meaning
Classical / high temperature	$\hbar\omega\ll k_BT:\ S(\omega)=\dfrac{2k_BT}{\omega}\chi''(\omega)$	$\coth x\approx1/x$ ; recovers Nyquist and Green–Kubo forms.
Quantum / zero temperature	$T\to0:\ S(\omega)=2\hbar\,\Theta(\omega)\chi''(\omega)$	Only spontaneous (zero-point + emission) noise survives; absorption noise is suppressed — vacuum fluctuations.
Static / zero frequency	$\omega\to0:\ \chi(0)=\beta\langle\delta A\,\delta B\rangle$	Static susceptibility equals the equilibrium covariance over $k_BT$ — the compressibility/susceptibility sum rule.
Single Brownian particle	$D=\mu k_BT,\ \mu=1/\gamma$	Velocity-correlation form at zero frequency: mobility $\mu$ and diffusion $D$ tied by $k_BT$ (Einstein).

Common trapOnly the dissipative part, only in equilibrium

The FDT relates the dissipative (imaginary, absorptive) part $\chi''$ to the symmetrized fluctuation spectrum — not the full $\chi$ to the raw correlation, and not the reactive $\chi'$ at all. Two errors are common: forgetting the $\coth(\hbar\omega/2k_BT)$ in the quantum case (which discards zero-point noise and gives $S\to0$ at $T=0$ , wrongly), and applying the theorem out of equilibrium — in a driven steady state, an aging glass, or active matter — where it simply does not hold. The second "error" is actually a tool: the measured ratio $\omega S/(2\chi'')$ departing from $k_BT$ is precisely how nonequilibrium is detected and quantified.

Worked examples

Example 1 — Johnson noise of a real resistor. This computes the actual voltage you would measure across a laboratory resistor from the FDT, a number an instrument engineer must know.

Step 1. Take $R=1\,\mathrm{k}\Omega$ at $T=300\,\mathrm{K}$ measured over a bandwidth $\Delta f=10\,\mathrm{kHz}$ , with $k_B=1.381\times10^{-23}\,\mathrm{J/K}$ .

Step 2. Apply the Nyquist form $\langle V^2\rangle=4k_BTR\,\Delta f=4\,(1.381\times10^{-23})(300)(10^3)(10^4)$ .

Step 3. Evaluate the product: $4\times1.381\times10^{-23}\times300=1.657\times10^{-20}$ ; times $10^3\times10^4=10^7$ gives $\langle V^2\rangle=1.657\times10^{-13}\,\mathrm{V^2}$ .

Step 4. Take the root: $\sqrt{\langle V^2\rangle}=\sqrt{1.657\times10^{-13}}\approx4.07\times10^{-7}\,\mathrm{V}$ . The resistor hisses at about $0.41\,\mu\mathrm{V}$ RMS — the fundamental thermal-noise floor no amplifier design can beat without lowering $T$ or $R$ .

Example 2 — The Einstein relation from the zero-frequency FDT. This shows the diffusion–mobility relation is the single-particle, static special case of the theorem, with a concrete number.

Step 1. For a tagged particle the velocity response to a constant force is the mobility $\mu$ (drift $=\mu F$ ), the $\omega\to0$ response; the diffusion constant is $D=\int_0^\infty\langle v(t)v(0)\rangle\mathrm{d}t$ , the zero-frequency velocity correlation.

Step 2. The FDT at $\omega=0$ ties correlation to response: $D=\mu k_BT$ . Take a micron-scale bead in water with mobility $\mu=1/\gamma$ where the Stokes drag gives $\gamma=6\pi\eta a$ for $a=1\,\mu\mathrm{m}$ , $\eta=1.0\times10^{-3}\,\mathrm{Pa\cdot s}$ : $\gamma=6\pi(10^{-3})(10^{-6})\approx1.885\times10^{-8}\,\mathrm{kg/s}$ .

Step 3. Then $\mu=1/\gamma=5.31\times10^{7}\,\mathrm{s/kg}$ and $D=\mu k_BT=(5.31\times10^{7})(1.381\times10^{-23})(300)$ .

Step 4. Evaluate: $D=(5.31\times10^{7})(4.143\times10^{-21})\approx2.20\times10^{-13}\,\mathrm{m^2/s}$ . The bead diffuses at $D\approx0.22\,\mu\mathrm{m^2/s}$ , fixed entirely by its drag and the temperature — fluctuation ( $D$ ) and dissipation ( $\gamma$ ) welded by $k_BT$ .

Origins · 1 minryogo kubo and how this was discovered

ryogo kubo

1920–1995, Japanese theoretical physicist · Ryogo Kubo reorganized nonequilibrium statistical mechanics around the correlation function. In his landmark 1957 paper he showed that every linear transport coefficient and susceptibility is a retarded equilibrium average of a commutator — the Kubo formula — and thereby made the fluctuation-dissipation theorem a corollary of linear response rather than a collection of special cases. His two-volume treatise with Toda and Hashitsume became the standard reference, and the "Kubo formula" and "Green–Kubo relations" now name the entire framework. Kubo did not discover the FDT — Einstein, Nyquist, Callen and Welton preceded him — but he gave it its general, calculable form, turning a family of noise formulas into a single equilibrium identity.

History

1905

— Einstein relates diffusion to mobility,

D=\mu k_BT

1928

— Johnson measures thermal resistor noise; Nyquist derives

\langle V^2\rangle=4k_BTR\,\Delta f

the same year

1931

— Onsager's regression hypothesis gives the time-domain statement

1951

— Callen and Welton prove the general quantum fluctuation-dissipation theorem

1957

— Kubo recasts the FDT as a corollary of linear response

1966

— Kubo's review formalizes the fluctuation-dissipation framework

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., ch. 15 (Fluctuations and Linear Response) · Kardar, Statistical Physics of Particles, ch. 9 · Reif, Fundamentals of Statistical and Thermal Physics, ch. 15 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, ch. 10 · Landau & Lifshitz, Statistical Physics Part 1, §118–§124 · H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951) · R. Kubo, Rep. Prog. Phys. 29, 255 (1966)

Provenance. Equations and limits verified against Pathria & Beale ch. 15, Kardar SPP ch. 9, Reif ch. 15, and Landau–Lifshitz Stat. Phys. Part 1 §118–§124. A verifier recomputes: (1) the classical FDT by-parts derivation yielding $S=(2k_BT/\omega)\chi''$ ; (2) the KMS detailed-balance relation $S(-\omega)=e^{-\beta\hbar\omega}S(\omega)$ and its assembly into $\coth(\hbar\omega/2k_BT)$ (Callen–Welton, Phys. Rev. 83, 34); (3) the Nyquist estimate $\approx0.41\,\mu$ V for $1\,\mathrm{k}\Omega$ , $300$ K, $10$ kHz using $k_B=1.381\times10^{-23}\,\mathrm{J/K}$ ; (4) the Langevin noise amplitude $\Gamma=2\gamma k_BT$ from equipartition and the Einstein $D=\mu k_BT$ Stokes-bead number. No nonstandard values asserted.

Builds on: Linear Response Theory · Brownian Motion · The Langevin Equation · Wiener Process and White Noise. Leads to: The Einstein Relation · The Ornstein-Uhlenbeck Process · The Fokker-Planck Equation · Kramers' Escape Rate Theory.

Why is

\chi''

and not the full response

\chi

tied to the fluctuations?

Because only $\chi''$ , the imaginary/dissipative part, describes irreversible energy uptake from a periodic drive — power absorbed is $\propto\omega\chi''(\omega)|f|^2$ . Energy balance in equilibrium says whatever the system absorbs it must, on average, give back as fluctuations, so only the dissipative channel appears in the theorem. The reactive part $\chi'$ describes lossless, in-phase energy storage and is not fed by the fluctuations directly. It is not independent, though: causality (Kramers–Kronig) determines $\chi'$ from $\chi''$ , so measuring the fluctuation spectrum ultimately fixes the entire response, reactive part included.

What is the difference between the classical and quantum forms, physically?

The classical form $S=(2k_BT/\omega)\chi''$ says fluctuations vanish as $T\to0$ : no thermal energy, no jiggle. The quantum form replaces $k_BT$ by $\tfrac{\hbar\omega}{2}\coth(\hbar\omega/2k_BT)$ , the mean energy of a harmonic oscillator including its ground-state $\tfrac12\hbar\omega$ . As a result the quantum spectrum does not vanish at $T=0$ — zero-point motion keeps the system fluctuating. The crossover happens at $\hbar\omega\sim k_BT$ : below it the mode is thermally excited and classical; above it the mode is frozen into its ground state and only quantum fluctuations remain. This is why the quantum correction matters for high-frequency modes and low-temperature devices.

How is the second fluctuation-dissipation theorem related to the first?

The "first" FDT is the frequency-domain statement $S(\omega)=(2k_BT/\omega)\chi''(\omega)$ relating an observable's fluctuation spectrum to its dissipative response. The "second" FDT is the noise-amplitude condition on a Langevin equation, $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$ , tying the random-force strength to the friction. They are the same physics at two levels of description: the second FDT is what you must impose on the microscopic Langevin model so that it reproduces the first FDT (and thermalizes to Maxwell–Boltzmann). If you set the noise amplitude to anything other than $2\gamma k_BT$ , the particle equilibrates to the wrong temperature and the response–fluctuation relation breaks.

Does the theorem require the system to be linear?

The response side is linear by construction — the theorem relates first-order response to equilibrium fluctuations. The fluctuations themselves need not be Gaussian or the dynamics linear; the correlation function $C(t)=\langle A(t)A(0)\rangle$ is a genuine equilibrium quantity of the full interacting system. What is required is that the reference state be true thermal equilibrium and that the perturbation be weak enough that only the linear response matters ( $|Bf|\ll k_BT$ ). For strong driving, higher-order response functions $\chi^{(2)},\chi^{(3)},\dots$ appear, and there are nonlinear generalizations of the FDT that relate them to higher equilibrium correlations, but the clean two-object identity is a linear-response statement.

How does the FDT actually detect that a system is out of equilibrium?

Measure two things independently: the response $\chi''(\omega)$ (drive the system weakly and record the absorptive part) and the fluctuation spectrum $S(\omega)$ (record spontaneous fluctuations with no drive). In equilibrium the ratio $\omega S(\omega)/2\chi''(\omega)$ must equal $k_BT$ at every frequency. If instead the ratio is frequency-dependent — say larger at low frequencies in an aging glass — the system is not in equilibrium, and the ratio defines a frequency-dependent effective temperature $T_{\rm eff}(\omega)$ . This "FDT-violation" measurement is the standard experimental probe for glasses (slow structural modes look hotter than the bath) and active matter (molecular-motor activity injects extra low-frequency fluctuations).

Why does the Green–Kubo formula follow from the FDT?

Transport coefficients are zero-frequency limits of dissipative responses. The FDT relates that dissipation to the fluctuation spectrum, and the fluctuation spectrum at zero frequency is the time-integral of the equilibrium autocorrelation of the relevant flux. Put together, a transport coefficient equals $\int_0^\infty\langle J(0)J(t)\rangle\,\mathrm{d}t$ up to a $k_BT$ factor — the Green–Kubo relation. So the same theorem that gives Johnson noise for a resistor gives the diffusion constant from the velocity autocorrelation and the viscosity from the stress autocorrelation. The Kubo formula develops these current-correlation expressions in full; here we note they are FDT corollaries at $\omega\to0$ .