Fluctuation-Dissipation Relation Near Criticality

The fluctuation-dissipation relation is one of the deepest organizing principles of statistical mechanics: a system's spontaneous equilibrium fluctuations and its linear response to a weak external perturbation are governed by the same underlying correlations. For a uniform magnet the static version says that the magnetic susceptibility — a response quantity measuring how much magnetization is induced per unit field — equals (up to a factor $k_BT$ ) the volume integral of the connected magnetization correlation function. This is exact in equilibrium and follows directly from differentiating the partition function twice with respect to the field. Near a critical point this identity becomes spectacular. The correlation function develops a diverging correlation length $\xi$ , so spins become correlated over macroscopic distances. The integral of the connected correlator $G_c(r)$ then grows without bound, and the susceptibility diverges. Fluctuations and response do not merely both diverge — they diverge in lockstep, with their ratio pinned by temperature. The relation therefore converts statements about correlations (the geometry of fluctuations) into statements about thermodynamic response, and vice versa. Feeding the scaling form of the correlation function into the relation yields one of the cornerstone scaling relations, $\gamma=\nu(2-\eta)$ , connecting the susceptibility exponent $\gamma$ to the correlation-length exponent $\nu$ and the anomalous dimension $\eta$ . This is a hallmark of universality: distinct microscopic systems sharing dimensionality and symmetry obey the same relation with the same exponents. The relation is thus the bridge between the correlation function and the critical exponents, and it generalizes — through the Kubo formula and Onsager regression — to dynamics, where it produces critical slowing down with a dynamic exponent $z$ . The relation is firmly an equilibrium statement. Out of equilibrium — in driven steady states, glasses, or systems aging after a quench through $T_c$ — the temperature-controlled proportionality between fluctuation and response breaks down, and the deviation (encoded in an effective temperature or a fluctuation-dissipation ratio) becomes a diagnostic of how far the system sits from equilibrium.

Key takeaways

The static fluctuation-dissipation relation for a magnet states that the susceptibility $\chi$ is the spatial integral of the connected spin-spin correlation function: $k_BT\,\chi=\int\mathrm{d}^d r\,G_c(r)$ .
Equivalently $\chi$ is the variance of the total order parameter, $\chi=\beta\,\mathrm{Var}(M)/V$ , and it equals the $q\to0$ structure factor $\tilde G_c(0)$ — measured directly by scattering.
Near a continuous transition the correlation length $\xi$ diverges, so the integrated fluctuations and hence $\chi$ diverge together with the same singularity $\chi\sim|t|^{-\gamma}$ .
Inserting the scaling form $G_c(r)\sim r^{-(d-2+\eta)}g(r/\xi)$ into the relation forces Fisher's identity $\gamma=\nu(2-\eta)$ .
The dynamic generalization ties the response function to the time-correlation function and yields critical slowing down, $\tau\sim\xi^{\,z}$ ; the relation is an equilibrium identity and fails out of equilibrium.
Key idea: In equilibrium the susceptibility IS the integrated connected correlation function, $k_BT\chi=\int\mathrm{d}^d r\,G_c(r)$ ; at criticality both sides diverge as $\xi\to\infty$ , and matching their scaling forms yields Fisher's relation $\gamma=\nu(2-\eta)$ .

The core insight

In equilibrium the susceptibility is the integrated connected correlation function, $k_BT\chi=\int\mathrm{d}^d r\,G_c(r)$ ; at criticality both sides diverge as $\xi\to\infty$ , and matching the scaling forms gives Fisher's relation $\gamma=\nu(2-\eta)$ .

Where this lives today

The static structure factor $S(q)=k_BT\chi/(1+q^2\xi^2)$ is measured every day at synchrotron and neutron sources: small-angle scattering off critical binary fluids, polymer blends near their consolute point, and alloys near order-disorder transitions all show the forward-scattering peak growing and narrowing as $T\to T_c$ , a direct readout of the fluctuation-dissipation relation. In numerical statistical mechanics the variance form $\chi=\beta V(\langle m^2\rangle-\langle m\rangle^2)$ with finite-size scaling $\chi\sim L^{\gamma/\nu}$ is the standard route to critical exponents. Beyond equilibrium, the fluctuation-dissipation ratio has become a working definition of temperature for glasses, granular media, and active matter, where it quantifies how far a system sits from equilibrium.

The physical picture

Poke a magnet with a tiny uniform field. How strongly it responds — its susceptibility — depends on how coordinated the spins already are: if flipping one spin drags a whole correlated patch along, the response is enormous. The size of that patch is the correlation volume $\sim\xi^d$ , and the relation says the response is proportional to that volume of fluctuation. Near $T_c$ the patches grow without bound, so a feeble field recruits a diverging number of spins and the susceptibility blows up. Fluctuation and response are two views of the same correlated cloud, so they diverge together with the same exponent. The algebra behind this is a single coincidence. Treat the partition function as a generating function: one field derivative gives the mean magnetization, a second gives the susceptibility — and that second derivative is also the variance of the total magnetization. The same object is both "response" and "fluctuation." That is the whole content of the relation, and it is exact.

Figure

Response equals the volume of the correlated patch — small ξ off-critical, diverging ξ at T_c

Out of equilibrium the link snaps. In a glass slowly settling, the spontaneous fluctuations report a high "effective temperature" while the measured response reflects the real bath temperature; the ratio of the two becomes a thermometer for how far from equilibrium the system sits.

The core equations

The central identity is that the static uniform susceptibility equals the integrated connected correlation function. It follows exactly from a double field-derivative of the partition function and holds at any temperature; its power appears near $T_c$ , where the right side diverges.

k_BT\,\chi \;=\; \int \mathrm{d}^d r\; G_c(\mathbf{r}),\qquad G_c(\mathbf{r})=\langle s(\mathbf{0})s(\mathbf{r})\rangle-\langle s\rangle^2 \qquad (1)

where: $\chi=\partial m/\partial h|_{h=0}$ is the static uniform susceptibility; $G_c(\mathbf r)$ is the connected (cumulant) spin-spin correlation function; $s(\mathbf r)$ is the local order-parameter density; $k_BT$ sets the fluctuation-response ratio; $d$ is the spatial dimension. Physically: the response is the total amount of correlation a spin has with the rest of the system. The same second derivative that produced $\chi$ is a variance, so the susceptibility is literally the fluctuation of the extensive order parameter. This is the form used to measure exponents in simulations.

\chi \;=\; \frac{\beta}{V}\,\mathrm{Var}(M)=\frac{\beta}{V}\big(\langle M^2\rangle-\langle M\rangle^2\big),\qquad M=\sum_i s_i \qquad (2)

where: $M$ is the total (extensive) magnetization; $V$ the number of sites/volume; $\beta=1/k_BT$ . Physically: no field derivative is needed — just watch how much $M$ wanders. In Fourier space the integrated correlator is the $q\to0$ limit of the structure factor, which is exactly what a scattering experiment reads off at forward angle.

\tilde G_c(\mathbf q)=\int\mathrm{d}^d r\, e^{-i\mathbf q\cdot\mathbf r}G_c(\mathbf r),\qquad k_BT\,\chi=\tilde G_c(\mathbf q\to 0) \qquad (3)

where: $\tilde G_c(\mathbf q)$ is the static structure factor; $\mathbf q$ the wavevector; $q\to0$ is the uniform (long-wavelength) response. Near $T_c$ the correlator takes a universal scaling form controlled by the diverging correlation length and two exponents, the anomalous dimension $\eta$ and the correlation-length exponent $\nu$ .

G_c(\mathbf r)\;\sim\; \frac{1}{r^{\,d-2+\eta}}\,g\!\left(\frac{r}{\xi}\right),\qquad \xi\sim|t|^{-\nu} \qquad (4)

where: $\eta$ is the anomalous (Fisher) dimension; $\xi$ the correlation length; $g(x)\to$ const as $x\to0$ and decays for $x\gg1$ ; $t=(T-T_c)/T_c$ . Physically: at $T_c$ correlations are pure power law (scale-free); away from it they are cut off at $\xi$ . Combining (1) and (4) forces a relation among the exponents — Fisher's scaling relation — which is the punchline of the whole article.

\gamma \;=\; \nu\,(2-\eta),\qquad \chi\sim|t|^{-\gamma} \qquad (5)

where: $\gamma$ is the susceptibility exponent. Physically: the divergence of the response is dictated by the geometry of the correlations, not independent of it. The dynamic generalization relates the linear response function to the time derivative of the autocorrelation function, the classical Kubo form.

R(t-t')=\frac{1}{k_BT}\,\Theta(t-t')\,\Big(\!-\frac{\partial}{\partial t}\,C(t-t')\Big) \qquad (6)

where: $R$ is the response function $\delta\langle s(t)\rangle/\delta h(t')$ ; $C(t-t')=\langle s(t)s(t')\rangle_c$ the autocorrelation; $\Theta$ enforces causality. As $\xi\to\infty$ the relaxation time diverges, $\tau\sim\xi^{\,z}$ — critical slowing down.

Derivation AStatic fluctuation-dissipation relation from the partition function

The goal is to show, with no approximation, that the static susceptibility equals the integrated connected correlation function, exhibiting the susceptibility as the second cumulant of the free energy with respect to a source field. We add a uniform field coupling linearly to the order parameter, differentiate the log-partition function twice, and use translation invariance to collapse the double lattice sum into a single spatial integral. This is the exact equilibrium statement whose divergence at $T_c$ is the physics of the article.

Step 1. Add a source: $Z(h)=\mathrm{Tr}\,e^{-\beta(\mathcal H_0-hM)}$ with $M=\sum_i s_i$ , so $-\beta F(h)=\ln Z(h)$ . Why: coupling $h$ linearly to $M$ makes field-derivatives pull down factors of $M$ .

Step 2. First derivative gives the mean order parameter, $\langle M\rangle=\beta^{-1}\,\partial_h\ln Z$ . Why: this is the generating-function property of $\ln Z$ — the first cumulant.

Step 3. Second derivative gives the variance, $\partial_h\langle M\rangle=\beta\big(\langle M^2\rangle-\langle M\rangle^2\big)$ . Why: differentiating a first cumulant of a Boltzmann average returns the second cumulant times $\beta$ .

Step 4. Expand the variance in two-point functions: $\langle M^2\rangle-\langle M\rangle^2=\sum_{i,j}G_c(\mathbf r_i-\mathbf r_j)=V\sum_j G_c(\mathbf r_j)$ . Why: translation invariance makes the double sum depend only on the separation, giving one factor of $V$ .

Step 5. Divide by $V$ for the intensive susceptibility: $\chi=V^{-1}\partial_h\langle M\rangle=\beta\sum_j G_c(\mathbf r_j)\to\beta\int\mathrm d^d r\,G_c(\mathbf r)$ , i.e. $k_BT\chi=\tilde G_c(q\to0)$ . Why: the continuum limit turns the site sum into the $q=0$ structure factor. *Verified: verify_aq-sm-fluctuation-dissipation-criticality.py checks that for non-interacting spins ( $G_c(r)=\delta_{r,0}$ ) both forms give $\chi=\beta$ , and that the variance and integral forms agree numerically on a 1D Ising chain.*

Go deeper · Derivation B · 1 minFisher's relation γ = ν(2−η) from the scaling correlator

Derivation BFisher's relation γ = ν(2−η) from the scaling correlator

The goal is to derive the exponent identity that ties the susceptibility divergence to the correlation-length divergence and the anomalous dimension. We insert the universal scaling form of the correlation function into the exact fluctuation-dissipation relation, rescale the integration variable by the correlation length, and count powers of $\xi$ . The residual integral is a pure convergent number, so the entire $|t|$ -dependence of the susceptibility comes from the explicit powers of $\xi$ — reading them off gives Fisher's relation, a relation obeyed by every system in a given universality class.

Step 1. Start from the exact relation $k_BT\chi=\int\mathrm d^d r\,G_c(\mathbf r)$ and substitute the scaling form $G_c(r)=r^{-(d-2+\eta)}g(r/\xi)$ . Why: near $T_c$ this ansatz captures the correlator's shape with $\xi$ the only scale.

Step 2. Rescale $u=r/\xi$ : the measure gives $\xi^d$ , the prefactor $r^{-(d-2+\eta)}=\xi^{-(d-2+\eta)}u^{-(d-2+\eta)}$ , so the net power of $\xi$ is $d-(d-2+\eta)=2-\eta$ . Why: this is dimensional counting — the correlation length is the only length, so all $r$ -dependence collapses onto it.

Step 3. The remaining integral $\int\mathrm d^d u\,u^{-(d-2+\eta)}g(u)$ is a finite constant for $0\le\eta<2$ , so $k_BT\chi\propto\xi^{\,2-\eta}$ . Why: $g(u)$ cuts off the large- $u$ tail and $u^{-(d-2+\eta)}$ is integrable at the origin in $d$ dimensions when $\eta<2$ .

Step 4. Insert $\xi\sim|t|^{-\nu}$ to get $\chi\sim|t|^{-\nu(2-\eta)}$ , and compare with $\chi\sim|t|^{-\gamma}$ to read off $\gamma=\nu(2-\eta)$ . Why: the exponent is what survives; the non-universal amplitude is irrelevant to it. *Verified: verify_aq-sm-fluctuation-dissipation-criticality.py confirms that the 2D Ising triple $(\nu,\eta)=(1,\tfrac14)$ gives $\gamma=\tfrac74$ and the mean-field triple $(\nu,\eta)=(\tfrac12,0)$ gives $\gamma=1$ , and that the rescaling power equals $2-\eta$ symbolically.*

Go deeper · Derivation C · 1 minClassical dynamic fluctuation-dissipation theorem and critical slowing down

Derivation CClassical dynamic fluctuation-dissipation theorem and critical slowing down

The goal is to extend the static relation to time-dependent response, linking the dissipative part of the dynamic susceptibility to the noise spectrum and showing why relaxation freezes at $T_c$ . We use Onsager's regression hypothesis — a spontaneous fluctuation decays by the same law as a macroscopically prepared disturbance — together with stationarity of the equilibrium time series, then take the frequency-domain imaginary part. The zero-frequency limit recovers the static relation, closing the loop, and the divergence of the relaxation time as $\xi\to\infty$ is critical slowing down.

Step 1. Onsager regression: $R(\tau)=-\,k_BT^{-1}\,\Theta(\tau)\,\dot C(\tau)$ , with $C(\tau)=\langle s(\tau)s(0)\rangle_c$ . Why: in equilibrium the decay of a fluctuation and of an imposed perturbation are the same process.

Step 2. Fourier transform over $\tau>0$ : $\chi(\omega)=\int_0^\infty\!\mathrm d\tau\,e^{i\omega\tau}R(\tau)$ . Why: causality restricts $R$ to positive times, defining the retarded response.

Step 3. Take the imaginary part and use stationarity $C(\tau)=C(-\tau)$ : $\chi''(\omega)=\dfrac{\omega}{2k_BT}\,S(\omega)$ with $S(\omega)=\int\mathrm d\tau\,e^{i\omega\tau}C(\tau)$ . Why: integrating by parts converts $\dot C$ into a factor $\omega$ acting on the symmetric spectrum $S$ .

Step 4. With $\tau\sim\xi^{\,z}\sim|t|^{-\nu z}$ , the dissipation peak sits at $\omega\sim\tau^{-1}\to0$ as $\xi\to\infty$ . Why: the relaxing mode becomes infinitely slow, so all spectral weight collapses to zero frequency. *Verified: verify_aq-sm-fluctuation-dissipation-criticality.py checks the classical fluctuation-dissipation prefactor $\omega/2k_BT$ on an Ornstein-Uhlenbeck process (Lorentzian $C$ and $\chi''$ ), and that the $\omega\to0$ limit reproduces the static $\chi=\int_0^\infty R\,\mathrm d\tau$ .*

Limiting cases

Case	Result	Meaning
Far from $T_c$ (short range)	$\xi$ finite, $k_BT\chi=\int\mathrm d^d r\,G_c(r)<\infty$	Correlations decay exponentially, $G_c\sim e^{-r/\xi}$ ; the integral converges and $\chi$ is finite and analytic — ordinary linear response.
Approaching $T_c$	$\chi\sim\|t\|^{-\gamma}$ , $\xi\sim\|t\|^{-\nu}$	Both diverge; matching through the relation forces $\gamma=\nu(2-\eta)$ .
Exactly at $T_c$	$G_c(r)\sim r^{-(d-2+\eta)}$ , $\tilde G_c(q)\sim q^{-(2-\eta)}$	Scale-free power-law correlations; the $q\to0$ limit is the divergent susceptibility.
Ordered phase, continuous symmetry	$\chi_\perp\to\infty$ for $T<T_c$ , $d\le4$	Transverse (Goldstone) modes are massless, so the transverse susceptibility diverges everywhere below $T_c$ .
Out of equilibrium / aging	$R\neq k_BT^{-1}\,\partial_{t'}C$	The relation is violated; the ratio $X=k_BT\,R/\partial_{t'}C\neq1$ defines an effective temperature $T_{\mathrm{eff}}=T/X$ .

Worked examples

Example 1 — Fisher's relation as a numerical consistency check. Verify that the exact 2D Ising exponents satisfy $\gamma=\nu(2-\eta)$ , and compare with mean field.

Step 1. For the 2D Ising universality class the exact values are $\nu=1$ and $\eta=\tfrac14$ . Compute $\gamma=\nu(2-\eta)=1\cdot(2-0.25)=1.75$ . The relation predicts $\gamma=\tfrac74$ , exactly the Onsager value.

Step 2. For mean field ( $d\ge4$ ): $\nu=\tfrac12$ , $\eta=0$ , so $\gamma=\tfrac12\cdot2=1$ — the Curie-Weiss result $\gamma=1$ .

Step 3. For 3D Ising, $\nu\approx0.630$ , $\eta\approx0.036$ : $\gamma\approx0.630\cdot(2-0.036)=1.237$ , matching the tabulated $\gamma\approx1.237$ . The single relation reproduces three universality classes.

Example 2 — extracting $\gamma/\nu$ from a simulation at $T_c$ . In a Monte Carlo run one measures the order-parameter variance rather than a field derivative. Show how finite-size scaling of $\chi$ yields $\gamma/\nu=2-\eta$ .

Step 1. At $T_c$ the correlation length is cut off by the box, $\xi\to L$ , so $\chi\sim\xi^{\,2-\eta}\to L^{\,2-\eta}$ . Since $\gamma/\nu=2-\eta$ , this is $\chi\sim L^{\gamma/\nu}$ .

Step 2. For 3D Ising, $2-\eta=2-0.036=1.964$ , so $\chi(L)\sim L^{1.964}$ . Doubling the box, $L\to2L$ , multiplies $\chi$ by $2^{1.964}=3.90$ . A measured factor of about $3.9$ per doubling confirms $\gamma/\nu\approx1.96$ — the fluctuation-dissipation relation turned into an estimator, no field derivative required.

Origins · 1 minstatic structure factor and how this was discovered

static structure factor

the q→0 fluctuation observable · The static structure factor S(q)=∫G_c(r)e^{−iq·r}d^d r is the Fourier transform of the connected correlation function and the quantity a scattering experiment measures. Its q→0 limit is the fluctuation-dissipation relation itself: S(0)=k_BT χ. Near a critical point S(q) takes the Ornstein–Zernike Lorentzian form S(q)=S_0/(1+q²ξ²), so the forward-scattering peak height tracks χ∼

History

1905

— Einstein relates diffusion to mobility (D = μ k_BT), the first fluctuation-dissipation statement

1928

— Nyquist and Johnson relate thermal voltage noise to resistance, the electrical fluctuation-dissipation law

1931

— Onsager states the regression hypothesis, that fluctuations decay by macroscopic laws

1944

— Onsager's exact 2D Ising solution fixes ν = 1 and (later) η = 1/4

1951

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

1957

— Kubo formulates linear response and the Green–Kubo relations

1964

— Fisher establishes the scaling relation γ = ν(2−η) linking susceptibility and correlations

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 12–13 · Kardar, Statistical Physics of Fields, Ch. 2–4 · 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, §116, §146–147 · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 7.

Provenance. The static relation is derived from $\partial^2\ln Z/\partial h^2$ (Pathria & Beale Ch. 12; Reif Ch. 15; Kardar SPF Ch. 2). Fisher's relation $\gamma=\nu(2-\eta)$ is checked against the scaling relations in Kardar SPF Ch. 4 and Goldenfeld Ch. 7. Exact exponent values rechecked: 2D Ising $\nu=1,\eta=\tfrac14,\gamma=\tfrac74$ (Onsager / McCoy–Wu); mean-field $\nu=\tfrac12,\eta=0,\gamma=1$ ; 3D Ising $\nu\approx0.630,\eta\approx0.036,\gamma\approx1.237$ . The dynamic theorem and Onsager regression follow Landau–Lifshitz SP1 §118 and Chaikin–Lubensky.

Builds on: Correlation Function · Critical Exponents · Continuous Phase Transitions · Order Parameter. Leads to: Scaling Hypothesis and Scaling Relations · Universality and Universality Classes · Goldstone Modes.

Is the fluctuation-dissipation relation the same as the fluctuation-dissipation theorem?

They are the static and dynamic faces of one idea. The static relation used here, $k_BT\chi=\int G_c$ , is the zero-frequency statement: response equals integrated equilibrium fluctuation. The dynamic theorem (Callen–Welton, Kubo) relates the full frequency-dependent dissipative response $\chi''(\omega)$ to the noise spectrum $S(\omega)$ . Setting $\omega=0$ in the dynamic theorem recovers the static relation. In this article "relation" is used for the static, thermodynamic identity that connects susceptibility to correlations near criticality.

Why do fluctuations and response diverge with the same exponent rather than two different ones?

Because they are literally the same object. The second field-derivative of $\ln Z$ is simultaneously the susceptibility (a response) and the variance of the order parameter (a fluctuation). There is no room for two exponents: one quantity, one singularity. This is why measuring the variance of $M$ in a simulation directly yields the susceptibility exponent, and why scattering (a fluctuation probe) and thermodynamic measurements (a response probe) agree.

What exactly is the anomalous dimension η, and why does it appear here?

At $T_c$ the correlation function decays as a power law $G_c(r)\sim r^{-(d-2+\eta)}$ . Naive (Gaussian/Ornstein–Zernike) theory predicts the exponent $d-2$ ; the extra $\eta$ is the anomalous correction from interactions, small but nonzero below the upper critical dimension. It enters Fisher's relation because the integral $\int G_c$ scales as $\xi^{2-\eta}$ , so $\eta$ directly modifies how fast the susceptibility diverges relative to the correlation length. In mean field $\eta=0$ ; in 3D Ising $\eta\approx0.036$ ; in 2D Ising $\eta=\tfrac14$ .

How can I measure the susceptibility exponent in a Monte Carlo simulation?

Use the variance form, not a field derivative. Compute $\chi=\beta V(\langle m^2\rangle-\langle m\rangle^2)$ at each temperature. At $T_c$ the correlation length is capped by the system size, so $\chi\sim L^{\gamma/\nu}=L^{2-\eta}$ ; running several box sizes and fitting the slope on a log-log plot gives $\gamma/\nu$ . Combined with an independent measurement of $\nu$ (from the temperature-derivative of the Binder cumulant, say), this delivers $\gamma$ . The fluctuation-dissipation relation is what makes the cheap variance measurement equal to the expensive response.

Does the relation hold in the ordered phase below T_c?

Yes, but you must be careful about which susceptibility. The longitudinal susceptibility (response along the order) stays finite away from $T_c$ . For a system with a continuous broken symmetry, the transverse susceptibility diverges everywhere in the ordered phase for $d\le4$ , because the Goldstone modes are massless and their fluctuations integrate to infinity — a direct fluctuation-dissipation consequence. For a discrete (Ising) symmetry there are no Goldstone modes and this transverse divergence is absent.

What is critical slowing down and how is it related to this relation?

As $T\to T_c$ the correlation length diverges, and the dynamic version of the relation ties the relaxation time to it: $\tau\sim\xi^{\,z}$ with a dynamic exponent $z$ . Physically, ordering a region of size $\xi$ takes a time that grows with $\xi$ , so the system relaxes ever more slowly and the dissipation spectrum $\chi''(\omega)$ piles up at $\omega\to0$ . This is why experiments near critical points require long equilibration and why simulations suffer from correlated samples — the same physics, seen through the dynamic fluctuation-dissipation theorem.