Johnson-Nyquist Noise: Why Every Resistor Hisses at Temperature T

What Johnson-Nyquist noise actually is

Inside any resistor the conduction electrons are in constant, chaotic thermal motion — a gas of charges jostling at roughly 10⁵ m/s of thermal speed, colliding with the lattice every ~10^-14 s. At any instant slightly more electrons happen to be drifting toward one terminal than the other, so a tiny, randomly fluctuating voltage appears across the ends. Averaged over time it is zero; its variance is not.

Two properties make it special:

• It is universal. The mean-square voltage V² = 4 k_B T R Δf contains no material constants — no mobility, no carrier density, no geometry. A carbon film, a metal foil, and a salt solution of the same R and T hiss identically.
• It is an equilibrium effect. Unlike shot noise, it needs no applied voltage or current. A resistor sitting in a drawer is emitting it right now.

Here k_B = 1.380649×10^-23 J/K is Boltzmann's constant, T the absolute temperature in kelvin, R the resistance in ohms, and Δf the measurement bandwidth in hertz. The voltage spectral density S_V = 4 k_B T R is flat — the noise is white across all practical frequencies.

The mechanism and Nyquist's derivation

J. B. Johnson measured the effect at Bell Labs in 1928, finding a voltage independent of the resistor's composition but proportional to R and T. His colleague Harry Nyquist explained it the same year with a beautifully economical thermodynamic argument.

Nyquist imagined two equal resistors R connected by a lossless transmission line of length L, the whole system in a thermal bath at temperature T. Each resistor delivers noise power into the line; in equilibrium the power flowing each way must be equal. Treating the line's electromagnetic modes as one-dimensional oscillators and assigning each the classical equipartition energy k_BT per mode, he counted the modes in bandwidth Δf and found the available noise power delivered to a matched load:

• P = k_B T Δf (maximum power into a matched load — note it is independent of R).

Converting available power back to an open-circuit voltage across R (a matched load sees V/2 across itself) gives the mean-square EMF V² = 4 k_B T R Δf. This is the earliest concrete instance of the fluctuation-dissipation theorem (Callen and Welton, 1951): the same electron scattering that dissipates energy as resistance also generates the fluctuating EMF, with strength fixed by T.

Key quantities and a worked example

The most useful engineering form is the voltage spectral density e_n = √(4 k_B T R), quoted in nV/√Hz. Work it out for a 1 kΩ resistor at 290 K (17 °C, the standard reference temperature):

• 4 k_B T R = 4 × 1.381×10^-23 × 290 × 1000 ≈ 1.60×10^-17 V²/Hz
• e_n = √(1.60×10^-17) ≈ 4.00 nV/√Hz (≈ 4.07 nV/√Hz at 300 K)

Because it scales as √R, a handy rule is 0.13×√R nV/√Hz at room temperature: a 1 MΩ resistor gives ~127 nV/√Hz, a 50 Ω line gives ~0.9 nV/√Hz. Over a 1 MHz bandwidth the 1 kΩ resistor produces V_rms = 4.0 nV/√Hz × √(10⁶) = 4 µV. The equivalent noise current is I_n = √(4 k_B T Δf / R) (Norton form), and the available power kTB in a 1 Hz band is 4×10^-21 W — i.e. -174 dBm/Hz, the number every RF engineer memorizes as the thermal noise floor.

How it's observed, measured, and put to work

Johnson noise is not merely a nuisance; it is a resource and a ruler:

• Primary thermometry. Because V²/Δf = 4 k_B R T links a measured voltage to T through fundamental constants only, Johnson Noise Thermometry (JNT) gives an absolute, drift-free temperature. NIST and NMIs used cross-correlated JNT in the 2017 redetermination of k_B that fed into the 2019 SI redefinition of the kelvin.
• The receiver noise floor. Every antenna, LNA, and ADC inherits kTB. Engineers quote a device's noise figure and noise temperature relative to a 290 K resistor; a receiver's sensitivity can never beat the -174 dBm/Hz thermal floor without cooling.
• Calibration standards. A resistor at a known T is a traceable noise source.

Measurement is subtle: the signal is microvolts of white noise buried under amplifier noise and pickup. Practitioners use cross-correlation of two independent amplifier chains reading the same resistor, so uncorrelated amplifier noise averages away as 1/√(measurement time) while the correlated Johnson signal survives. Cooling the resistor (T → 4 K or below) is the direct way to suppress it, which is why sensitive front-ends and quantum experiments live in dilution refrigerators.

How it differs from shot, flicker, and quantum noise

Thermal noise is easy to confuse with its neighbors, but the distinctions are sharp:

• vs. shot noise (Schottky, 1918): shot noise, S_I = 2qI, arises from the discreteness of charge crossing a barrier and requires net current flow; Johnson noise needs none. A diode in reverse bias hisses with shot noise; an unbiased resistor hisses with Johnson noise.
• vs. flicker (1/f) noise: flicker noise rises without limit toward low frequency and does depend on material, defects, and bias. Johnson noise is flat and material-independent. Below a device's 1/f corner frequency, flicker dominates; above it, Johnson noise wins.
• vs. quantum/zero-point: Nyquist's classical k_BT-per-mode breaks down when hf ≈ k_BT. The full quantum formula replaces k_BT with hf/(e^hf/k_BT−1) + ½ hf, adding a zero-point term. At 300 K the crossover is f = k_BT/h ≈ 6.25 THz, far above ordinary electronics — which is why the noise looks perfectly white in practice.

Significance, subtleties, and open questions

Johnson-Nyquist noise is the archetype linking irreversibility (resistance dissipates) to fluctuation (the same channel injects random EMF). That link, formalized as the fluctuation-dissipation theorem, now spans Brownian motion, dielectric loss, and gravitational-wave detector thermal noise — the very floor that limits LIGO's mirror suspensions.

• The zero-point controversy. Whether the ½ hf zero-point term produces real, measurable voltage fluctuations in a resistor, or is an artifact of the detection scheme, has been debated for decades and remains a live discussion in the metrology and mesoscopics literature.
• Beyond equilibrium. In driven or quantum-coherent conductors, the neat 4kTR fails; quantum shot noise and Fano-factor measurements now probe fractional charge (e.g. e/3 quasiparticles in the fractional quantum Hall effect) and non-equilibrium fluctuation relations.
• Practical frontier. Squeezing, parametric amplification, and cryogenics push detectors below the naive kTB floor, and JNT is a candidate for a fully electronic, calibration-free temperature standard.

A century on, the humble hiss of a resistor still marks the boundary between what physics lets us measure and what it forever hides.

Johnson-Nyquist noise versus its close cousins in electronics

Noise type	Physical origin	Spectrum	Governing formula	Needs current flow?
Johnson-Nyquist (thermal)	Thermal agitation of carriers in equilibrium	White (flat to ~THz)	S_V = 4 k_B T R	No
Shot noise	Discreteness of charge crossing a barrier	White	S_I = 2 q I	Yes
Flicker (1/f) noise	Trapping/defect fluctuations, surfaces	Rises as 1/f at low f	S_V ∝ 1/f	Usually yes
Generation-recombination	Carrier number fluctuation in semiconductors	Lorentzian (rolls off)	S ∝ τ/(1+(2πfτ)²)	Yes
Quantum / zero-point	Vacuum fluctuations at hf ≫ k_B T	Rises linearly with f	S_V = 2 h f R	No