fluctuation-dissipation theoremtheorem
The theorem was rigorously formulated by Herbert Callen and Theodore Welton in 1951 on the basis of quantum statistical mechanics. However, its origins go back to Albert Einstein's 1905 work on Brownian motion, where he related the mobility of a particle (dissipative response) to its diffusion (fluctuations). Even earlier, in 1928, Harry Nyquist derived a formula for noise in resistors: mean square voltage ⟨V²⟩ = 4k_B T R Δf, where Δf is the bandwidth. The Callen–Welton theorem generalizes this connection to any systems near equilibrium: the spectrum of spontaneous fluctuations is proportional to the imaginary part of the susceptibility.
How it works
In everyday life: in headphones, a quiet noise is heard even without a signal — this is the thermal noise of electrons in wires, predicted by the theorem. The higher the speaker's resistance, the louder the hiss. That is why sensitive audio equipment is made from low-resistance materials or is cooled.
💡 Electrical noise set a sensitivity limit for radio receivers: even an ideal amplifier cannot be quieter than thermal noise, which is critical for communication with distant spacecraft.
S(\omega) = \frac{\hbar}{\pi} \coth\left(\frac{\hbar\omega}{2k_B T}\right) \operatorname{Im}\chi(\omega)
S(ω) — spectral density of fluctuations of the observable; ω — angular frequency; ħ — reduced Planck constant (ħ ≈ 1.0546×10⁻³⁴ J·s); k_B — Boltzmann constant (k_B ≈ 1.380649×10⁻²³ J/K); T — absolute temperature; χ(ω) — dynamic susceptibility (response function); Im — imaginary part.
\langle V^2 \rangle = 4 k_B T R \Delta f
⟨V²⟩ — mean square noise voltage; k_B — Boltzmann constant (≈ 1.380649×10⁻²³ J/K); T — temperature; R — electrical resistance; Δf — measurement bandwidth.