---
title: Linear Response and the Fluctuation-Dissipation Theorem
module: Fluctuations and Response
moduleNumber: 12
lessonNumber: 3
order: 1203
summary: >
  A system driven by a weak external field responds through a generalized
  susceptibility $\chi(\omega)$ whose imaginary part measures dissipation. The
  Wiener–Khinchin theorem makes the power spectrum of equilibrium fluctuations the
  Fourier transform of their correlation function, and the fluctuation–dissipation
  theorem ties the two together: $S_x(\omega)=(2k_BT/\omega)\,\chi''(\omega)$, so
  the spectrum of spontaneous fluctuations is fixed by the dissipative response.
  The Johnson–Nyquist noise of a resistor, $\langle V^2\rangle=4k_BTR\,\Delta f$,
  is the canonical example, and Onsager reciprocity closes the subject.
topics: [Fluctuations and Response]
sources:
  - book: Kubo
    ref: "The fluctuation-dissipation theorem (Rep. Prog. Phys. 29, 255, 1966)"
  - book: Kardar (Statistical Physics of Particles)
    ref: "Ch. 9 — nonequilibrium; linear response and the fluctuation–dissipation theorem, §9.1–9.4"
  - book: Reif
    ref: "Ch. 15 — Irreversible Processes and Fluctuations; §15.8–15.15"
  - book: Pathria & Beale
    ref: "Ch. 13 — Fluctuations and Nonequilibrium Statistical Mechanics; §13.4–13.6"
draft: false
---

The [Einstein relation](/statistical-mechanics/fluctuations/brownian-motion-and-the-langevin-equation)
$D=\mu_{\mathrm{mob}}k_BT$ and the Langevin noise
$\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\,\delta(t-t')$ each tie a fluctuation to a
dissipation for one specific system. The fluctuation–dissipation theorem is the
statement that this tie is universal and dynamical: for any system near
equilibrium, the spectrum of spontaneous fluctuations of a quantity is fixed by
the dissipative part of that quantity's response to a weak external field. The
same molecular collisions that let a system absorb energy from a drive also
generate its equilibrium fluctuations, and the theorem gives the exact
proportionality at every frequency. This lesson develops the linear-response
formalism, defines the correlation functions and their power spectra, states the
Wiener–Khinchin theorem, and assembles them into the fluctuation–dissipation
theorem. The Johnson–Nyquist noise of a resistor is the worked example, and
Onsager's reciprocal relations follow from the same time-reversal symmetry that
underlies the theorem.

## Linear response and the response function

Perturb a system by coupling an observable $A$ to a weak, time-dependent external
field $f(t)$, adding $-f(t)A$ to the Hamiltonian. To first order in $f$ the
deviation of $\langle A\rangle$ from its equilibrium value is linear in the field,
and by time-translation invariance it depends only on the elapsed time:

$$
\delta\langle A(t)\rangle=\int_{-\infty}^{\infty}\chi(t-t')\,f(t')\,\d t'.
$$

The kernel $\chi(\tau)$ is the **response function**. Causality — the response
cannot precede the cause — forces $\chi(\tau)=0$ for $\tau<0$, so the integral runs
only over $t'<t$. The response function is an equilibrium property of the
unperturbed system; the field only reads it out.

> **Definition (Response function and generalized susceptibility).** The response
> function $\chi(\tau)$ relates the linear response to a field through
> $\delta\langle A(t)\rangle=\int_{-\infty}^{t}\chi(t-t')f(t')\,\d t'$. Its Fourier
> transform is the **generalized susceptibility**
> $$\chi(\omega)=\int_0^\infty\chi(\tau)\,e^{i\omega\tau}\,\d\tau=\chi'(\omega)+i\chi''(\omega),$$
> with real part $\chi'$ (the in-phase, reactive response) and imaginary part
> $\chi''$ (the out-of-phase, dissipative response).

Under a monochromatic drive $f(t)=f_0\cos\omega t$ the steady response is
$\delta\langle A(t)\rangle=f_0\,[\chi'\cos\omega t+\chi''\sin\omega t]$: the
in-phase part follows the drive, the quadrature part lags it by a quarter cycle.
The lag is what carries energy from the field into the system, so $\chi''$ is the
dissipative response. Causality relates $\chi'$ and $\chi''$ through the
Kramers–Kronig dispersion relations, integral transforms of each other, so a
system that dissipates at some frequencies must respond reactively at others; the
two parts are not independent.[^kk]

$$
% caption: A weak monochromatic drive produces a response of the same frequency but reduced amplitude and a phase lag; the in-phase amplitude is $\chi'(\omega)$ and the quarter-cycle lagging component is $\chi''(\omega)$, the dissipative part that absorbs power from the field.
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$$

## Correlation functions and the power spectrum

In equilibrium the field is off, but $A$ still fluctuates. Its fluctuations are
characterized by the **autocorrelation function**

$$
C(\tau)=\langle\delta A(0)\,\delta A(\tau)\rangle,
\qquad
\delta A=A-\langle A\rangle,
$$

which is stationary (independent of the time origin) and even in $\tau$ for a
classical variable. It starts at the variance $C(0)=\langle\delta A^2\rangle$ and
decays to zero over a correlation time $\tau_c$ as the fluctuation loses memory.
The spectral content of the fluctuations is the **power spectral density**, the
Fourier transform of the correlation function:

$$
S_A(\omega)=\int_{-\infty}^{\infty}C(\tau)\,e^{i\omega\tau}\,\d\tau.
$$

> **Theorem (Wiener–Khinchin).** For a stationary process the power spectral
> density and the autocorrelation function are a Fourier-transform pair,
> $$
> S_A(\omega)=\int_{-\infty}^{\infty}\langle\delta A(0)\,\delta A(\tau)\rangle\,e^{i\omega\tau}\,\d\tau,
> \qquad
> \langle\delta A(0)\,\delta A(\tau)\rangle=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_A(\omega)\,e^{-i\omega\tau}\,\d\omega.
> $$
> Setting $\tau=0$ gives the total variance as the integrated spectrum,
> $\langle\delta A^2\rangle=(2\pi)^{-1}\!\int S_A(\omega)\,\d\omega$.

The theorem converts a measurement in the time domain (a correlation) into one in
the frequency domain (a spectrum) and back. A fluctuation that decays as
$C(\tau)=\langle\delta A^2\rangle\,e^{-\lvert\tau\rvert/\tau_c}$ has the Lorentzian
spectrum $S_A(\omega)=\langle\delta A^2\rangle\,2\tau_c/(1+\omega^2\tau_c^2)$: white
at low frequency and rolling off above $1/\tau_c$. The area under the spectrum is
the variance regardless of the line shape.

$$
% caption: The Wiener–Khinchin theorem: an exponentially decaying autocorrelation $C(\tau)=\langle\delta A^2\rangle e^{-|\tau|/\tau_c}$ (left) transforms into a Lorentzian power spectrum $S_A(\omega)=\langle\delta A^2\rangle\,2\tau_c/(1+\omega^2\tau_c^2)$ (right), flat below $1/\tau_c$ and falling above it.
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$$

## The fluctuation-dissipation theorem

The response function and the correlation function are not independent. Both are
built from the same equilibrium dynamics of $A$, and the Kubo formula expresses
$\chi(\tau)$ as an equilibrium average involving the time derivative of the
correlation function. Fourier transforming that relation yields the central
result.

> **Theorem (Fluctuation–dissipation).** The power spectrum of the equilibrium
> fluctuations of $A$ is fixed by the dissipative part of its susceptibility,
> $$S_A(\omega)=\frac{2k_BT}{\omega}\,\chi''(\omega)\qquad(\hbar\omega\ll k_BT).$$
> The full quantum form replaces the prefactor,
> $$S_A(\omega)=\hbar\coth\!\left(\frac{\hbar\omega}{2k_BT}\right)\chi''(\omega),$$
> which reduces to the classical expression when $\hbar\omega\ll k_BT$ and to the
> zero-point spectrum $S_A=\hbar\,\chi''$ when $\hbar\omega\gg k_BT$.

The spectrum of spontaneous fluctuations at frequency $\omega$ is proportional to
how strongly the system dissipates energy at that frequency. A system that absorbs
readily at $\omega$ also fluctuates strongly at $\omega$, and $k_BT$ sets the
scale. Integrating the classical form over all frequencies recovers the static
identity of the [first lesson](/statistical-mechanics/fluctuations/thermodynamic-fluctuations-and-response):
$\langle\delta A^2\rangle=(2\pi)^{-1}\!\int S_A\,\d\omega=k_BT\,\chi'(0)=k_BT\,\chi_T$,
so the equilibrium variance equals $k_BT$ times the static susceptibility, exactly
$\langle\Delta A^2\rangle=k_BT\,(\partial\langle A\rangle/\partial f)_T$. The
dynamical theorem contains the static one as its zero-frequency integral.

The energy the system absorbs from a monochromatic drive confirms that $\chi''$ is
dissipation. Averaging the power $f(t)\,\d\langle A\rangle/\d t$ over a cycle of
$f(t)=f_0\cos\omega t$ gives

$$
\overline{P}=\tfrac12\,\omega\,\chi''(\omega)\,f_0^2\;\ge\;0.
$$

Positivity of the dissipated power requires $\omega\,\chi''(\omega)\ge0$: the
dissipative response has the same sign as the frequency. Through the
fluctuation–dissipation theorem this is the same statement as $S_A(\omega)\ge0$, a
power spectrum being non-negative.

$$
% caption: The dissipative response $\chi''(\omega)$ peaks where the system absorbs energy most readily; the fluctuation–dissipation theorem makes the equilibrium fluctuation spectrum $S_A(\omega)=(2k_BT/\omega)\chi''(\omega)$ proportional to it, so the absorption line and the noise spectrum carry the same information.
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$$

## Johnson–Nyquist thermal noise

An electrical resistor at temperature $T$ generates a fluctuating voltage across
its terminals even with no current source attached, because the thermal motion of
its charge carriers produces a randomly varying charge imbalance. This is
**Johnson–Nyquist noise**, and the fluctuation–dissipation theorem fixes its
spectrum. Take the observable to be the charge, driven by an applied voltage. The
response of the current to the voltage is the admittance $Y(\omega)$, whose
dissipative (real) part for an ideal resistor is $\operatorname{Re}Y=1/R$,
independent of frequency. Applying the theorem to the current and converting to the
voltage across the resistor gives a white noise spectrum.

> **Theorem (Nyquist).** The mean-square voltage fluctuation of a resistor $R$ at
> temperature $T$ in a measurement bandwidth $\Delta f$ is
> $$\langle V^2\rangle=4k_BTR\,\Delta f,$$
> corresponding to a one-sided voltage-noise spectral density $S_V=4k_BTR$,
> independent of frequency until $\hbar\omega\sim k_BT$.

The noise is white because the dissipation $\operatorname{Re}Y=1/R$ is
frequency-independent; the quantum factor $\coth(\hbar\omega/2k_BT)$ rolls the
spectrum off only at frequencies where $\hbar\omega$ approaches $k_BT$, far above
the range of ordinary electronics. The result contains no property of the resistor
beyond its resistance and temperature: the same $R$ that dissipates power $I^2R$
when a current flows generates the voltage noise when none does. Measuring the
noise is an absolute thermometer, and the value of $k_B$ obtained this way agrees
with every other determination.[^nyquist]

$$
% caption: A resistor at temperature $T$ produces a white voltage-noise spectrum $S_V=4k_BTR$, flat because the dissipation $\mathrm{Re}\,Y=1/R$ is frequency-independent; the spectrum rolls off only near $\hbar\omega\sim k_BT$, where the quantum factor $\coth(\hbar\omega/2k_BT)$ departs from its classical value.
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$$

## Onsager reciprocity

When several fluxes respond to several driving forces, the linear coefficients form
a matrix: a flux $J_i$ (of heat, charge, or particles) responds to the whole set of
thermodynamic forces $X_j$ (gradients of temperature, potential, or concentration)
through

$$
J_i=\sum_j L_{ij}\,X_j.
$$

The diagonal coefficients are the ordinary transport coefficients — thermal
conductivity, electrical conductivity, diffusivity — and the off-diagonal ones are
the cross effects, such as thermoelectricity, where a temperature gradient drives a
charge current and a voltage drives a heat current. Onsager showed that the matrix
is symmetric.

> **Theorem (Onsager reciprocity).** For fluxes and forces defined from the
> entropy production, the linear transport coefficients are symmetric,
> $$L_{ij}=L_{ji},$$
> in the absence of a magnetic field or overall rotation (which reverse under time
> reversal, giving $L_{ij}(\vec B)=L_{ji}(-\vec B)$).

The symmetry follows from the same principle behind the fluctuation–dissipation
theorem: microscopic reversibility. Equilibrium correlation functions are
invariant under time reversal, $\langle\delta A_i(0)\,\delta A_j(t)\rangle=\langle\delta A_i(t)\,\delta A_j(0)\rangle$,
because the microscopic equations of motion run equally well forward and backward.
The transport coefficients are time integrals of these correlations (Green–Kubo
relations), so the symmetry of the correlations becomes the symmetry of the
coefficients. Onsager reciprocity is the macroscopic shadow of time-reversal
symmetry, exactly as the fluctuation–dissipation theorem is its dynamical
expression at each frequency.[^onsager]

$$
% caption: The linear transport matrix $L_{ij}$ couples each flux to every force; the diagonal entries are the direct transport coefficients and the off-diagonal entries the cross effects, which Onsager reciprocity makes equal, $L_{12}=L_{21}$, from time-reversal symmetry.
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$$

## Summary

- Linear response to a weak field $f$ coupled to $A$ is
  $\delta\langle A(t)\rangle=\int\chi(t-t')f(t')\d t'$ with $\chi(\tau)=0$ for
  $\tau<0$; its transform is the generalized susceptibility
  $\chi(\omega)=\chi'+i\chi''$, whose imaginary part $\chi''$ is the dissipative,
  quarter-cycle-lagging response.
- The **Wiener–Khinchin theorem** makes the power spectrum $S_A(\omega)$ the
  Fourier transform of the autocorrelation $C(\tau)=\langle\delta A(0)\delta A(\tau)\rangle$;
  the integrated spectrum is the variance.
- The **fluctuation–dissipation theorem** $S_A(\omega)=(2k_BT/\omega)\chi''(\omega)$
  (classical), with quantum factor $\hbar\coth(\hbar\omega/2k_BT)$, fixes the
  equilibrium fluctuation spectrum by the dissipative response; its
  zero-frequency integral is the static identity $\langle\delta A^2\rangle=k_BT\chi_T$.
- The average dissipated power under a drive is
  $\overline{P}=\tfrac12\omega\chi''f_0^2\ge0$, so $\omega\chi''\ge0$ and the noise
  spectrum is non-negative.
- **Johnson–Nyquist noise** $\langle V^2\rangle=4k_BTR\,\Delta f$ is the theorem
  applied to a resistor, whose frequency-independent dissipation $\operatorname{Re}Y=1/R$
  makes the noise white until $\hbar\omega\sim k_BT$.
- **Onsager reciprocity** $L_{ij}=L_{ji}$ follows from the time-reversal symmetry
  of equilibrium correlations, the same microscopic reversibility that underlies
  the fluctuation–dissipation theorem.

[^kk]: **Kubo**, "The fluctuation-dissipation theorem," _Reports on Progress in Physics_ **29**, 255 (1966), §2 — causality, the analytic structure of $\chi(\omega)$, and the Kramers–Kronig relations between $\chi'$ and $\chi''$.
[^nyquist]: **Reif**, _Fundamentals of Statistical and Thermal Physics_, §15.15, and **Pathria & Beale**, _Statistical Mechanics_ (4th ed.), §13.5 — the Nyquist theorem for thermal noise in a resistor and its derivation from the fluctuation–dissipation theorem.
[^onsager]: **Kubo**, _Reports on Progress in Physics_ **29**, 255 (1966), §5, and **Reif**, _Fundamentals of Statistical and Thermal Physics_, §15.8–15.10 — the Onsager reciprocal relations, the Green–Kubo formulas for transport coefficients, and their origin in the time-reversal symmetry of equilibrium correlation functions.