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Harry Nyquist was an electrical engineer whose work connected communication engineering with physical and mathematical limits. His 1928 theory explained thermal electrical fluctuations using thermodynamics and statistical mechanics, agreeing with John Bertrand Johnson's measurements.^[ref]

In the classical regime $hf\ll k_{B}T$, an approximately constant electrical resistance $R$ at temperature $T$ produces an open-circuit voltage expected value square $⟨V^2⟩=4k_{B}TR\Delta f$ over a one-sided frequency band of width $\Delta f$. Here $k_B$ is the Boltzmann constant. The formula connects Johnson-Nyquist noise with both the physical conductor and the signal bandwidth through which it is observed.^[ref]

By Harry Nyquist, published as H. Nyquist. Physical Review 32, 110–113; published 1 July 1928.

Cited passages: publisher abstract; p. 113, equations (4)–(8), inspected through indexed primary-text excerpt. This theoretical paper is distinct from Johnson’s measurements.

Thermal electrical noise is the fluctuating electric potential difference or electric current produced by thermal fluctuations in a dissipative electrical circuit, even without an applied voltage. A resistor at thermal equilibrium is both a source of noise and a sink for electrical energy. Its electrical resistance describes the dissipation, and its absolute temperature sets the scale of the fluctuations. This connects a practical limit on signal-to-noise ratio to statistical mechanics: a warm, unbiased resistor is not electrically silent.

John Bertrand Johnson measured the effect using a vacuum-tube amplifier and thermocouple; his 1928 paper reported agreement with the Boltzmann constant. Harry Nyquist's companion paper calculated the effect from thermodynamics and statistical mechanics.^[ref][ref]