Fluctuation-Dissipation Theorem

Equivalent forms

\langle F(t)F(0)\rangle = 2k_B T \gamma \delta(t) \quad\text{(white noise limit)}

S_V(f) = 4k_B T R \quad\text{(Johnson-Nyquist)}

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

One of the most profound results in physics: equilibrium fluctuation spectra are computable from equilibrium response functions — no need to model non-equilibrium dynamics.

Symbol	Name	SI unit	Dimension	Range
$S_FF(ω)$	Force noise power spectral densityoutput One-sided spectral density of thermal fluctuations	N²/Hz	ML²T⁻³	0 – 1e-20
$Re[χ(ω)]$	Real part of susceptibility (dissipative part) Imaginary part of the response function χ, governing energy absorption	m/N	M⁻¹T²	0 – 0.000001
$T$	Temperature Absolute temperature	K	Θ	1 – 1000
$R$	Resistance Electrical resistance (Johnson-Nyquist noise application)	Ω	ML²T⁻³A⁻²	1 – 1000000

Unit systems

SI:: S_V in $V^{2}/Hz = 4k_BT R$
electronics:: noise spectral density often quoted as $nV/\sqrt{H}z$ (square root)
quantum:: $\hbar = k_B = 1$ , response in natural units

Where it holds

Exact for linear response near equilibrium. Breaks down far from equilibrium, for nonlinear systems, or in driven open quantum systems.

Discovery

Herbert Nyquist / John Johnson / Herbert Callen & Theodore Welton · 1928

Johnson measured resistor noise in 1928; Nyquist derived it thermodynamically the same year. Callen and Welton gave the general quantum version in 1951, connecting noise to the imaginary part of any response function.

Try this

Why do resistors generate noise even when no current flows?

Johnson-Nyquist noise in a resistor comes from the same thermal fluctuations that dissipate energy when current flows. The fluctuation-dissipation theorem (FDT) unifies these phenomena: the spectrum of spontaneous fluctuations is directly related to the dissipative response to an external perturbation.

Research status: stable

Common misconceptions

FDT applies to LINEAR response only. Nonlinear susceptibilities (Raman scattering, harmonic generation) require higher-order fluctuation theorems.

Derivation

Define response

\chi (t) = \delta

⟨A(t)⟩/

\delta F(0)

.

Kubo formula:

\chi ''(\omega ) = (1/2\hbar )\int dt e^{i\omega t}

⟨[A(t),A(0)]⟩.

FDT:

S_AA(\omega ) = 2\hbar \chi ''(\omega )

coth

(\hbar \omega /2k_BT)

.

Classical limit: coth

(x) \to 1/x

for x ≪ 1, giving

S = 2k_BT \chi ''(\omega )

.

Limiting cases

Classical (k_BT ≫

\hbar \omega )

⟶

S(\omega ) = 2k_BT Re[\chi (\omega )]

— frequency-independent (white) noise

Quantum

(\hbar \omega

≫ k_BT)⟶

S(\omega ) = \hbar \omega Re[\chi (\omega )]

— zero-point quantum noise; survives at

T=0

Overdamped harmonic oscillator⟶

\chi (\omega ) = 1/(m\omega _{0}^{2} - m\omega ^{2} + i\gamma \omega )

; noise peaks at resonance

\omega _{0}

Boltzmann Entropy→Canonical Partition Function→