Temperature Is in the Jitter

The kernel, granted

Grant the accessible version first, because it is everywhere and correct. Nearly all temperature is measured electrically. A temperature-dependent electrical property stands in for the temperature, and you read the property instead of the heat.

The list is the furniture of every lab and thermostat. A thermocouple exploits the Seebeck effect: a temperature difference across junctions of two dissimilar conductors produces an EMF, the voltage proportional (to first order) to the difference (Seebeck, 1821; the effect was first seen by Volta in 1794 and named "thermoelectricity" by Ørsted). A platinum resistance thermometer (RTD) uses a pure-metal resistor whose resistance rises with temperature as lattice-phonon scattering grows — nearly linear, highly reproducible (Pt100: 100 Ω at 0 °C, IEC 60751). A thermistor is a metal-oxide semiconductor whose resistance falls steeply and non-linearly as it warms. A silicon sensor reads the temperature dependence of a junction voltage — the base-emitter drop and its proportional-to-absolute-temperature difference (the bandgap-reference circuit family, LM35 and kin).

Each is a real thermometer, and each is honestly a proxy: the relations are only approximately linear, and every one is calibrated against a reference. Seebeck coefficients drift with temperature; platinum follows the slightly quadratic Callendar–Van Dusen curve; the thermistor is exponential. These read temperature by comparison, through a fitted table. Real, ubiquitous — and secondary. They borrow their scale from somewhere.

The deep kernel

There is one that borrows from nowhere, and it is the elegant core of the seed.

Any resistor, sitting at temperature T, generates a fluctuating voltage across its own terminals — with no current driven through it, no battery, nothing applied. The electrons jostle with the heat, and their jostling is a small, noisy EMF. Johnson measured it (J. B. Johnson, "Thermal Agitation of Electricity in Conductors," Phys. Rev. 32, 97–109, 1928); Nyquist derived it in the paper immediately following, in the same issue (H. Nyquist, "Thermal Agitation of Electric Charge in Conductors," Phys. Rev. 32, 110–113, 1928). The mean-square noise voltage in a bandwidth Δf is

⟨V²⟩ = 4 k_B T R Δf

— Nyquist's result, four symbols and a bandwidth. Read the variance of the noise, know R and Δf, and you have T on the absolute scale directly. No reference thermometer, no fitted table, no calibration against a known temperature. This is a primary thermometer: the temperature falls straight out of the fluctuation. It is the textbook instance of the fluctuation–dissipation theorem, which Callen and Welton derived as a generalization of exactly this relation (H. B. Callen and T. A. Welton, "Irreversibility and Generalized Noise," Phys. Rev. 83, 34–40, 1951): the same resistance that dissipates energy also sets the size of the equilibrium fluctuations, and the fluctuations carry the temperature.

It is not a curiosity. Johnson noise thermometry was one of three techniques whose measurements fed the final determination of the Boltzmann constant for the 2019 redefinition of the kelvin, fixing k_B = 1.380649×10⁻²³ J K⁻¹ (NIST; CODATA 2017 adjustment; SI revision approved November 2018, in force 20 May 2019). It was the secondary contributor — acoustic gas thermometry dominated the weighted mean, with NIST's best noise result near 2.7 ppm — but it was there, reading temperature off a jiggle to help pin the unit of temperature itself.

The correction

Here is the spine, and it was in the seed from the start. That voltage ain't random.

Correct, and the whole claim turns on it. The noise across the resistor is the collective motion of some 10²³ electrons, each pushed and scattered under deterministic dynamics. The microstate at any instant is fixed — it is what it is, not a draw from a hat. Nothing about the wire is undecided. If you could track every charge and phonon, the voltage at the next instant would follow, in principle, with no dice thrown.

So "random" is not a property the voltage has. It is a coarse-graining verdict — a statement about our description, not about the wire. We do not track the 10²³ degrees of freedom; we track one number, the terminal voltage, and everything we declined to follow shows up in that number as jitter we cannot predict. This is the same move as the site's entropy paper: entropy is not a substance in the system but a number your description assigns. Randomness is the same kind of thing. The deterministic microdynamics only look random once you blur away the microstate you were never tracking — which is exactly what the grain instrument shows: blur a deterministic field and it reads as noise.

The correction cuts sharply once stated. Electricity as a signal is not a thermometer at all — a battery's steady voltage is fixed by its chemistry and says nothing about whether it is warm or cold. Raise its temperature and the terminal voltage barely stirs. What reads temperature is not the current and not the mean; it is the variance of the fluctuation around the mean. The jiggle, not the current. Two things set the fluctuation, and it is worth keeping them apart. Its shape is Gaussian because it is a central-limit sum over all those hidden, deterministic degrees of freedom — that is what the CLT delivers, the bell curve, not the size. Its magnitude — the reason it obeys 4 k_BT R Δf and not some arbitrary scatter — is fixed by the fluctuation–dissipation theorem, equipartition over the resistor's modes, which ties the size of the fluctuation to the dissipation. So treating the untracked microstate as a random sample of a thermal bath is not a guess: the CLT licenses the shape, the FDT licenses the size, and the treatment-as-random is exactly as principled as the physics it rests on.

The jiggle, not the current: temperature is read off the variance of a fluctuation that is deterministic underneath and random only in the description.

The precise claim

So the corrected seed, stated as a proposal: temperature is legible in electricity not as a value but as a spread. Not the voltage — the variance of the voltage's fluctuation. And that fluctuation is deterministic beneath and treated as random by necessity, because no one tracks 10²³ electrons.

The thermometer works because the untracked microstate can be treated as a thermal bath sample — and "random" and "temperature" are then read off the same coarse-grained description, in the same act. You blur to one number; the blurring is what makes the microstate look random; and the size of that apparent randomness is the temperature. They are not two facts about the wire. They are one description, read twice.

This ties in two directions. To the surround: a reading needs a reservoir, and Johnson noise makes it literal — the resistor reads the thermal bath it sits in, and the temperature in the formula is the bath's. And to entropy-exist: randomness, like entropy, is in the description, not in the thing.

The limits, plainly

The limits are where this stays honest, so state them.

• The formula has a quantum limit. 4 k_BT R Δf is the low-frequency, high-temperature form. The full Callen–Welton expression carries a Planck factor h f/(e^h f/k_BT−1) that suppresses the noise once h f approaches k_BT — a crossover near 6 THz at room temperature, far above ordinary electronics, but real at very high frequency or very low T. Whether the additive zero-point term is measurable as noise is a genuinely debated, convention-dependent question, not settled fact.
• Real measurements are not the clean formula. You need a defined bandwidth, a long averaging time to beat the variance down, and careful subtraction of amplifier noise. The primary thermometer is primary in principle; in practice it is delicate metrology.
• "Deterministic underneath" is a classical framing. The claim that the microstate is fixed and merely untracked is a statement about classical microdynamics. Quantum indeterminacy is a separate and genuinely open interpretive question, and this note does not resolve it — it does not claim the jiggle is deterministic all the way down, only that the classical-microdynamics reading is what licenses "random is in the description." Determinism "in principle" is also not tractability: no one integrates 10²³ electrons, and the untracked-ness is permanent, not a temporary gap.

There is a live instrument for this on the site: /field/jitter renders the fluctuation directly — the spread you can read a temperature off of.

Kin to Entropy Is in the Description (randomness, like entropy, is description-relative) and The Surround (a reading needs a reservoir — here, the thermal bath). The instrument: /field/jitter; the coarse-graining it rests on, /field/grain.

Rests on: Johnson, "Thermal Agitation of Electricity in Conductors" (Phys. Rev. 32, 97–109, 1928) and Nyquist, "Thermal Agitation of Electric Charge in Conductors" (Phys. Rev. 32, 110–113, 1928), giving ⟨V²⟩ = 4 k_BT R Δf; the fluctuation–dissipation theorem (Callen & Welton, Phys. Rev. 83, 34–40, 1951); Johnson noise thermometry's contribution to the Boltzmann-constant determination for the 2019 SI kelvin (NIST; CODATA 2017); the Seebeck effect (Seebeck, 1821), RTDs (IEC 60751), thermistors, and silicon junction sensors as secondary electrical thermometers. Noted as open: the quantum/zero-point correction beyond the classical form, and whether the microdynamics are deterministic beneath quantum indeterminacy. Reading temperature as the variance of a deterministic-but-untracked fluctuation — and "random" as a coarse-graining verdict rather than a property of the wire — is a proposal, offered to be argued with, not a proven identity, and not a challenge to the physics of any thermometer above.