Linear Response and the Fluctuation-Dissipation Theorem

The Einstein relation$D={\mu }_{\mathrm{mob}}{k}_{B}T$ and the Langevin noise $⟨\xi (t)\xi (t^{\prime })⟩=2\gamma k_{B}T\delta (t-t^{\prime })$ 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 $⟨A⟩$ from its equilibrium value is linear in the field, and by time-translation invariance it depends only on the elapsed time:

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 $t^{\prime }<t$. The response function is an equilibrium property of the unperturbed system; the field only reads it out.

Under a monochromatic drive $f(t)=f_0\cos \omega t$ the steady response is $\delta ⟨A(t)⟩=f_0[{\chi }^{\prime }\cos \omega t+{\chi }^{\prime \prime }\sin \omega t]$: 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 ${\chi }^{\prime \prime }$ is the dissipative response. Causality relates ${\chi }^{\prime }$ and ${\chi }^{\prime \prime }$ 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.¹

Correlation functions and the power spectrum

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

which is stationary (independent of the time origin) and even in $\tau$ for a classical variable. It starts at the variance $C(0)=⟨\delta A^2⟩$ 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:

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 $C(\tau )=⟨\delta A^2⟩e^{-|\tau |/{\tau }_c}$ has the Lorentzian spectrum $S_A(\omega )=⟨\delta A^2⟩2{\tau }_c/(1+{\omega }^2{\tau }_c^2)$: white at low frequency and rolling off above $1/{\tau }_c$. The area under the spectrum is the variance regardless of the line shape.

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.

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 $k_{B}T$ sets the scale. Integrating the classical form over all frequencies recovers the static identity of the first lesson: $⟨\delta A^2⟩=(2\pi {)}^{-1}\int S_A\mathbf{d}\omega =k_{B}T{\chi }^{\prime }(0)=k_{B}T{\chi }_T$, so the equilibrium variance equals $k_{B}T$ times the static susceptibility, exactly $⟨\Delta A^2⟩=k_{B}T(\partial ⟨A⟩/\partial f{)}_T$. The dynamical theorem contains the static one as its zero-frequency integral.

The energy the system absorbs from a monochromatic drive confirms that ${\chi }^{\prime \prime }$ is dissipation. Averaging the power $f(t)\mathbf{d}⟨A⟩/\mathbf{d}t$ over a cycle of $f(t)=f_0\cos \omega t$ gives

Positivity of the dissipated power requires $\omega {\chi }^{\prime \prime }(\omega )\ge 0$: the dissipative response has the same sign as the frequency. Through the fluctuation–dissipation theorem this is the same statement as $S_A(\omega )\ge 0$, a power spectrum being non-negative.

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 $\mathrm{Re}Y=1/R$, independent of frequency. Applying the theorem to the current and converting to the voltage across the resistor gives a white noise spectrum.

The noise is white because the dissipation $\mathrm{Re}Y=1/R$ is frequency-independent; the quantum factor $\coth (\hslash \omega /2k_{B}T)$ rolls the spectrum off only at frequencies where $\hslash \omega$ approaches $k_{B}T$, 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 $I^{2}R$ 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.²

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

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.

The symmetry follows from the same principle behind the fluctuation–dissipation theorem: microscopic reversibility. Equilibrium correlation functions are invariant under time reversal, $⟨\delta A_i(0)\delta A_j(t)⟩=⟨\delta A_i(t)\delta A_j(0)⟩$, 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.³

Summary

• Linear response to a weak field $f$ coupled to $A$ is $\delta ⟨A(t)⟩=\int \chi (t-t^{\prime })f(t^{\prime })\mathbf{d}{t}^{\prime }$ with $\chi (\tau )=0$ for $\tau <0$; its transform is the generalized susceptibility $\chi (\omega )={\chi }^{\prime }+i{\chi }^{\prime \prime }$, whose imaginary part ${\chi }^{\prime \prime }$ is the dissipative, quarter-cycle-lagging response.
• The Wiener–Khinchin theorem makes the power spectrum $S_A(\omega )$ the Fourier transform of the autocorrelation $C(\tau )=⟨\delta A(0)\delta A(\tau )⟩$; the integrated spectrum is the variance.
• The fluctuation–dissipation theorem $S_A(\omega )=(2k_{B}T/\omega ){\chi }^{\prime \prime }(\omega )$ (classical), with quantum factor $\hslash \coth (\hslash \omega /2k_{B}T)$, fixes the equilibrium fluctuation spectrum by the dissipative response; its zero-frequency integral is the static identity $⟨\delta A^2⟩=k_{B}T{\chi }_T$.
• The average dissipated power under a drive is $\overline{P}=\frac{1}{2}\omega {\chi }^{\prime \prime }{f}_0^2\ge 0$, so $\omega {\chi }^{\prime \prime }\ge 0$ and the noise spectrum is non-negative.
• Johnson–Nyquist noise $⟨V^2⟩=4k_{B}TR\Delta f$ is the theorem applied to a resistor, whose frequency-independent dissipation $\mathrm{Re}Y=1/R$ makes the noise white until $\hslash \omega \sim k_{B}T$.
• 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.