Fluctuation-Dissipation Theorem

13.1 Linear Response Theory

The fluctuation-dissipation theorem (FDT) connects the response of a system to a small perturbation with the spontaneous fluctuations of the system at equilibrium.

Consider a Hamiltonian $\mathcal{H}_0$ perturbed by a time-dependent field:

$\mathcal{H}(t) = \mathcal{H}_0 - f(t)A$

Where $A$ is an observable conjugate to the field $f(t)$ . The change in $\langle A(t) \rangle$ to first order in $f$ is:

$\langle A(t) \rangle - \langle A \rangle_0 = \int_{-\infty}^{t} \chi_{AA}(t - t")\, f(t')\, dt'$

Where the response function is:

$\chi_{AA}(t) = \frac{i}{\hbar}\theta(t)\langle[A(t), A(0)]\rangle_0$

13.2 Classical FDT

In the classical limit, the FDT takes a simpler form. The dynamic susceptibility $\chi(\omega) = \chi'(\omega) + i\chi''(\omega)$ relates to the power spectrum $S(\omega)$ of fluctuations:

$S(\omega) = \frac{2k_B T}{\omega}\,\chi''(\omega)$

For a harmonic oscillator with damping $\gamma$ and natural frequency $\omega_0$ :

$\chi''(\omega) = \frac{\gamma\omega}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}$

The fluctuation spectrum is Lorentzian, peaked at $\omega_0$ .

13.3 Johnson—Nyquist Noise

The FDT predicts thermal (Johnson—Nyquist) noise in a resistor:

$\langle V^2 \rangle = 4k_B T R \Delta f$

Where $R$ is the resistance and $\Delta f$ is the bandwidth. This noise is fundamental — it arises from thermal fluctuations of charge carriers and cannot be eliminated.

Worked Example 13.1: Johnson--Nyquist Noise Calculation

A $10$ k $\Omega$ resistor at room temperature ( $T = 300$ K) measured with bandwidth $\Delta f = 1$ MHz:

$\langle V^2 \rangle = 4 \times 1.38 \times 10^{-23} \times 300 \times 10^4 \times 10^6$

$= 4 \times 1.38 \times 10^{-23} \times 3 \times 10^{12}$

$= 1.66 \times 10^{-10} \text{ V}^2$

$V_{\text{rms} = \sqrt{1.66 \times 10^{-10}} \approx 1.29 \times 10^{-5} \text{ V} = 12.9 \text{ \mu\text{V}}}$

This sets a fundamental limit on the sensitivity of electrical measurements.

Worked Example 13.2: Brownian Motion and Einstein Relation

The Einstein relation is a special case of the FDT for Brownian motion. The diffusion constant $D$ relates to the mobility $\mu$ :

$D = \mu k_B T$

For a spherical particle of radius $r$ in a fluid with viscosity $\eta$ :

$\mu = \frac{1}{6\pi\eta r} \quad \text{(Stokes drag)}$

So $D = k_B T/(6\pi\eta r)$ .

For a $1$ $\mu$ M diameter sphere in water ( $\eta = 10^{-3}$ Pa $\cdot$ S) at $T = 300$ K:

$D = \frac{1.38 \times 10^{-23} \times 300}{6\pi \times 10^{-3} \times 0.5 \times 10^{-6}} = \frac{4.14 \times 10^{-21}}{9.42 \times 10^{-9}} \approx 4.39 \times 10^{-13} \text{ m}^2/\text{s}$

The mean squared displacement in time $t$ is $\langle x^2 \rangle = 2Dt$ . In 1 second: $\sqrt{\langle x^2 \rangle} \approx 0.94$ $\mu$ M.

Key Relationships

Quantity	Expression	Physical Content
Response function	$\chi_{AA}(t) = \frac{i}{\hbar}\theta(t)\langle[A(t), A(0)]\rangle_0$	Causal linear response
Classical FDT	$S(\omega) = \frac{2k_B T}{\omega}\chi''(\omega)$	Fluctuations $\leftrightarrow$ dissipation
Johnson—Nyquist	$\langle V^2 \rangle = 4k_B T R \Delta f$	Voltage noise in resistor
Einstein relation	$D = \mu k_B T$	Diffusion $\leftrightarrow$ mobility
Nyquist formula	$S_V(\omega) = 2k_B T R(\omega)$	Generalised impedance noise

Common Pitfalls

FDT assumes thermal equilibrium: The system must be in equilibrium at temperature $T$ . For out-of-equilibrium systems (driven, ageing, glasses), generalised fluctuation-dissipation relations apply with an effective temperature.
Quantum corrections at low temperature: The classical FDT $S(\omega) = 2k_B T \chi''(\omega)/\omega$ is valid only for $\hbar\omega \ll k_B T$ . At low temperatures, the quantum FDT gives $S(\omega) = \hbar\coth(\hbar\omega/2k_B T)\,\chi''(\omega)$ which includes zero-point fluctuations.
Response functions must be causal: The response function $\chi_{AA}(t)$ must vanish for $t < 0$ (causality). The Kramers—Kronig relations follow from this, connecting the real and imaginary parts of $\chi(\omega)$ .
Linearity requirement: FDT is a result of linear response theory. For large perturbations, nonlinear effects break the simple relation between fluctuations and dissipation.

Applications

Electrical engineering: Johnson—Nyquist noise sets the fundamental noise floor in amplifiers, sensors, and communication systems. Cryogenic cooling reduces thermal noise for sensitive measurements.
Brownian motion: The Einstein relation enables determining Boltzmann’s constant via tracking colloidal particles, or measuring Avogadro’s number from diffusion measurements.
Optical trapping: Fluctuations of a trapped bead in optical tweezers obey the FDT, allowing calibration of trap stiffness from the power spectrum of position fluctuations.
Gravitational wave detectors: Thermal noise in mirror suspensions and test masses limits the sensitivity of LIGO at intermediate frequencies.
Biophysics: Single-molecule force spectroscopy (optical tweezers, AFM) uses fluctuation analysis to extract spring constants and dissipation in biomolecules.

Connections to Other Topics

Kramers—Kronig relations: Causality imposes integral relations between $\chi'(\omega)$ and $\chi''(\omega)$ , which are intimately related to the FDT.
Onsager regression hypothesis: The relaxation of macroscopic nonequilibrium fluctuations follows the same laws as spontaneous fluctuations at equilibrium — a precursor to the FDT.
Nonequilibrium statistical mechanics: The fluctuation theorem (Evans—Searles, Crooks) extends fluctuation—dissipation ideas to far-from-equilibrium regimes, relating work distributions to free energy differences.
Quantum optics: The FDT applied to the electromagnetic field yields the Planck spectrum, connecting blackbody radiation to vacuum fluctuations and dissipation.

Summary Table: FDT in Different Contexts

System	Fluctuation	Dissipation	FDT Relation
Resistor	Voltage noise $\langle V^2 \rangle$	Resistance $R$	$\langle V^2 \rangle = 4k_B T R \Delta f$
Brownian particle	Position fluctuations $\langle x^2 \rangle$	Drag coefficient $\gamma$	$D = k_B T / \gamma$
Harmonic oscillator	Amplitude fluctuations	Damping rate $\Gamma$	$S_x(\omega) = \frac{2k_B T}{\omega}\,\chi''(\omega)$
Blackbody radiation	Field fluctuations	Absorption cross section	Planck spectrum $S(\omega) = \frac{\hbar\omega^3}{\pi^2 c^2}(e^{\hbar\omega/k_BT} - 1)^{-1}$
Magnetic system	Magnetisation noise	Magnetic susceptibility	$\chi''(\omega) = \frac{\omega}{2k_B T} S_M(\omega)$

Additional Worked Example: Fluctuation-Dissipation in an RLC Circuit

Problem. An RLC circuit at temperature $T$ has resistance $R$ , inductance $L$ , and capacitance $C$ . Find the power spectrum of voltage fluctuations across the capacitor using the FDT.

Solution. The impedance of the RLC circuit is $Z(\omega) = R + i(\omega L - 1/(\omega C))$ . The FDT in impedance form states: $\langle V^2 \rangle_\omega = 2k_B T \, \text{Re}[Z(\omega)]$ . For the RLC circuit:

$\langle V_C^2 \rangle_\omega = 2k_B T R \left| \frac{1/(i\omega C)}{R + i(\omega L - 1/(\omega C))} \right|^2$

The total mean-square voltage across the capacitor is $\langle V_C^2 \rangle = k_B T / C$ (equipartition). This is independent of $R$ , illustrating that the fluctuation-dissipation relation always yields the correct thermal equilibrium result regardless of the dissipation mechanism.

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