Illustrated Physics
physics, drawn
Thermodynamics

Half a kT in a piece of wire

Count the quadratic terms in a molecule's energy and equipartition gives a heat capacity. Count them on a transmission line instead and the same theorem gives a resistor's noise voltage — 4kTRΔf, with nothing in it about what the resistor is made of. A fifty-ohm input at room temperature says 0.91 nanovolts in every root hertz, and no design removes it.
17 min read 5 figures The same equation againWho is measuring

Assumes: Half a kT for every way of moving · The temperature a molecule does not have

Half a kT for every way of moving is a theorem about quadratic terms, and every use of it so far has been to matter — to the translations and rotations of a molecule, to the vibrations of a solid, to a chain of masses.

It is not a theorem about matter. It is a theorem about any system whose energy is a sum of quadratic terms in equilibrium at a temperature, and a piece of wire is such a system. Applying it there produces a number that is on the first page of every instrument’s specification and that no amount of good design removes.

The modes of a wire, counted. Nyquist's argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, and the line is a one-dimensional cavity whose standing waves are spaced c/2L apart in frequency. Each mode has an electric and a magnetic energy, both quadratic, so equipartition gives it kT — the same half a kT per quadratic term that a heat capacity counts. The upper points are how many modes fall in a band of a hundred and thirty-seven megahertz, for five line lengths, from 17 on a thirteen-metre line to 8,558 on one of six kilometres. The lower points are the power each end therefore receives, as a fraction of kTΔf, and they converge on one: a longer line has proportionally more modes and takes proportionally longer to deliver them, so the length cancels. The longest line lands within 0.005 per cent and the shortest is 4.5 per cent low, because thirteen metres holds only seventeen whole modes in the band and the remainder is a real granularity rather than an error. What is left in the limit is kTΔf — a noise power with no resistance in it at all, and none of the line's properties either.
Fig. 1 Nyquist’s argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, so the line is a one-dimensional cavity whose standing waves are spaced c/2L apart. Each mode has an electric and a magnetic energy, both quadratic, so each holds kT. The upper points are how many modes fall in a fixed band for five line lengths; the lower points are the power each end therefore receives, and the length cancels.

The argument, which is four sentences

Join two identical resistors by a lossless transmission line and wait. Everything is at one temperature, nothing is being driven, and the whole arrangement is in thermal equilibrium — which means each resistor is delivering exactly as much power to the line as it takes from it.

Now count what is in the line. It is a cavity of length LL, its standing waves are spaced c/2Lc/2L apart in frequency, and each of them has an electric energy and a magnetic energy, both quadratic in their amplitudes. Equipartition gives each quadratic term half a kBTk_BT, so each mode holds kBTk_BT — the same accounting that gives a diatomic gas its five halves and that fails for hydrogen’s rotations at low temperature for the same reason it will fail here at high frequency.

In a bandwidth Δf\Delta f there are 2LΔf/c2L\Delta f/c modes, holding 2LkBTΔf/c2Lk_BT\Delta f/c between them. All of that energy is travelling at cc, half in each direction, so it is all absorbed by the two resistors within one transit time L/cL/c. The power arriving at each is

P=2LkBTΔfc⋅c2L=kBTΔfP = \frac{2Lk_BT\Delta f}{c}\cdot\frac{c}{2L} = k_BT\Delta f

and the length has cancelled. So has the line’s impedance, its dielectric, and everything else about it. What is left is a power that the resistor must have been supplying, since in equilibrium it supplies what it absorbs, and a resistor supplying kBTΔfk_BT\Delta f into a matched load has an open-circuit noise voltage of 4kBTRΔf\sqrt{4k_BTR\Delta f}.

The figure performs the count, and the cancellation is visible as a convergence rather than an identity: a thirteen-metre line holds only seventeen whole modes in the band and is four per cent low, and a six-kilometre line holds eight and a half thousand and is within a twenty-thousandth. The granularity is real — a short line genuinely has a lumpy noise spectrum — and it disappears exactly as fast as the mode count grows.

Johnson measured the effect in 1927 and Nyquist explained it in 1928, in a paper of three pages with no adjustable constant in it.

What is not in the answer

The absences are the striking part, and they are worth listing because each of them was a surprise.

The material is not in it. Carbon, metal film, wirewound and a semiconductor channel of the same resistance at the same temperature produce the same noise. That is not an approximation; it is forced by the derivation, which never mentions what the resistor is.

The geometry is not in it. Nothing about the line survived, so nothing about the resistor’s shape can matter either.

And the resistance is not in the power. A large resistor produces a large voltage into a large impedance and delivers the same watts. The available noise power is kBTΔfk_BT\Delta f — 4.14×10−214.14\times10^{-21} watts per hertz at room temperature, which radio engineering writes as −174-174 dBm per hertz and puts at the top of every link budget.

Real resistors do differ, and the difference is instructive about what this essay covers. Pass a current through a carbon composition resistor and it produces an additional noise, proportional to the current and falling as one over the frequency, which depends entirely on how it is made. That is not thermal noise and it is not in equilibrium: it is a fluctuation in the resistance itself as current paths open and close, it vanishes with the current, and a resistor sitting at temperature with nothing through it has none of it.

An antenna, which is a resistor made of the sky

There is a consequence of the derivation’s indifference to what the resistor is that is worth a section, because it turns the result into an instrument.

An antenna has a resistance. Most of it is not ohmic: it is radiation resistance — the power the antenna delivers to the space around it, which is a loss as far as the circuit is concerned. By the argument above, an antenna in equilibrium with its surroundings must produce a noise voltage corresponding to that resistance and to the temperature of whatever it is radiating into.

So an antenna pointed at the ground, which is at 290 kelvin, is a 290-kelvin noise source. Pointed at the cold sky it is a much colder one — a few kelvin at centimetre wavelengths, because the sky at those wavelengths is nearly transparent and what is behind it is the microwave background. The antenna has not changed. What changed is the temperature of the thing its radiation resistance couples to.

That is why a radio astronomer quotes a system temperature rather than a noise voltage, and why the quantity is additive: the sky’s temperature, the ground’s leaking in through the sidelobes, the atmosphere’s, the feed’s own loss, the first amplifier’s. Each is a dissipation at a temperature, each contributes by the same relation, and the total is what limits the observation.

It is also how the microwave background was found. Penzias and Wilson had an antenna whose noise temperature came out three kelvin higher than everything they could account for, they eliminated every dissipation they could think of including the pigeons, and the residue was the temperature of the sky itself. A measurement of a noise voltage was a measurement of the temperature of the universe, and the chain of reasoning from one to the other is the argument in this essay’s first figure.

Where the count fails, and the same cure

Equipartition gives every mode kBTk_BT whatever its frequency, and there is no highest frequency, so the total noise power in the line is infinite.

The noise a wire does not have at high frequency. The mean energy of a mode of a wire, as a fraction of kT, against frequency on logarithmic axes, at three temperatures. Equipartition gives every mode kT regardless of its frequency, which is the flat line and which would make the total noise power infinite. What a mode actually holds is hf/(exp(hf/kT) − 1), which equals kT below the crossover — checked here three decades below it, where it agrees to two parts in a thousand — and falls exponentially above it. The crossover, found by bisection, is at 7.9e+12 Hz at 300 K, 2.0e+12 Hz at 77 K, 1.1e+11 Hz at 4.2 K. So the flat noise spectrum every engineer uses is the classical limit of a quantum expression, and it is flat over the whole of radio and microwave engineering because room temperature puts the crossover in the far infrared. This is the ultraviolet catastrophe arriving in a circuit, and it has the same cure: counting the modes was never the mistake, and giving each of them kT was.
Fig. 2 The mean energy of a mode of a wire, as a fraction of kT, against frequency, at three temperatures. Equipartition is the flat line, and it makes the total infinite. What a mode actually holds is hf/(exp(hf/kT) − 1), which equals kT below the crossover and falls exponentially above it. The bend is at six terahertz at room temperature and at ninety gigahertz at 4.2 kelvin.

That is the ultraviolet catastrophe, arriving in a circuit rather than in a cavity, and it has the same cure. A mode of frequency ff holds hf/(ehf/kBT−1)hf/(e^{hf/k_BT}-1) rather than kBTk_BT: the same below hf≈kBThf \approx k_BT, exponentially less above.

Counting the modes was never the mistake. Rayleigh and Jeans counted correctly; what was wrong was giving each mode kBTk_BT, which is the classical result and is the high-temperature limit of the quantum one. The identical correction fixes a solid’s heat capacity at low temperature, a diatomic gas’s rotations, and a wire’s noise spectrum, and it is one substitution in all three.

One curve for every solid, once the temperature is measured in its own units. The molar heat capacity of a solid in Debye's model, in units of the gas constant, against temperature divided by that solid's own Debye temperature. All four fall on one curve, which is the model's whole claim: a solid has one parameter and no others. It climbs to 3.000R, the Dulong and Petit value that every solid reaches when every mode is excited, and it falls at low temperature as the cube of the temperature with a coefficient of 233.782, both computed from the integral rather than quoted. The four solids reach half of Dulong and Petit at 26 K for lead, 85 K for copper, 160 K for silicon, 555 K for diamond — a spread of a factor of twenty, from one number each. The cube is the part the third law needs. Entropy is the integral of C/T from absolute zero, and an integrand going as T² converges there; a heat capacity that stayed at 3R all the way down would make that integral diverge logarithmically and there would be no absolute entropy to speak of.
Fig. 3 The same substitution, made on a solid: the heat capacity of a lattice against temperature, rising as the cube at low temperature because only the long-wavelength modes have enough energy to be excited, and flattening at the classical three R when they all do. A wire’s noise spectrum and a solid’s heat capacity are the same count of modes with the same correction applied, and the only difference is which axis the crossover is drawn against.

Where the bend sits decides whether any of it matters. At room temperature it is at six terahertz, which is beyond every circuit anybody builds, so the flat spectrum that engineers use is exactly right throughout radio, microwave and optical electronics. At 4.2 kelvin it is at ninety gigahertz — inside the band of a radio telescope’s receiver, which is one of the few instruments for which the classical formula is measurably wrong and which is designed with the quantum expression.

The numbers

What a resistor says when nothing is connected to it. The root-mean-square noise voltage across a resistor at room temperature, against its resistance over nine decades, for three measurement bandwidths — both axes logarithmic. The voltage is √(4kTRΔf), so it rises as the square root of the resistance and as the square root of the bandwidth: a 50 Ω radio input contributes 28.78 nanovolts in a kilohertz, a 1 kΩ source contributes 129 nanovolts in a kilohertz, a 1 MΩ oscilloscope contributes 4070 nanovolts in a kilohertz, a 10 GΩ electrometer contributes 407035 nanovolts in a kilohertz. A fifty-ohm input at room temperature gives 0.91 nanovolts in every root hertz, which is the number the whole of radio engineering is calibrated against. The expression contains the resistance, the temperature and the bandwidth and nothing else — not what the resistor is made of, not its shape, not whether it is carbon, wire or a channel of semiconductor. And the available power is kTΔf with no resistance in it at all, identical for every point on every line: a large resistor produces a larger voltage into a proportionally larger impedance, and delivers the same watts.
Fig. 4 The noise voltage across a resistor at room temperature against its resistance, over nine decades, for three bandwidths. A fifty-ohm input gives 0.91 nanovolts in every root hertz — the number radio engineering is calibrated against — and a ten-gigaohm electrometer gives twelve microvolts in a kilohertz. Every point on every line has the same available power behind it.

The square roots are what make the expression manageable and are also what make it hard to escape. Halving a noise voltage means quartering the resistance, quartering the bandwidth, or quartering the absolute temperature — and the last of those means going from room temperature to 75 kelvin, which is liquid nitrogen and a cryostat.

Which is why the design moves against thermal noise are so few and so consistent across quite different instruments. Lower the source resistance, which is why a photodiode amplifier’s feedback resistor is a compromise between gain and noise and why sensitive bridges are built at low impedance. Narrow the bandwidth, which is what a lock-in amplifier does and is the subject of the next figure. That move has a cost the figure makes explicit and a molecule’s own temperature fluctuation makes general: a narrower band is a longer measurement, and the quantity being measured has to hold still for it. Cool the source, which is why a radio telescope’s first amplifier is at twenty kelvin and why an infrared detector is at seventy-seven. There is no fourth move.

What averaging does, and where it stops

Narrowing the bandwidth by measuring for longer is the move available without changing any hardware, and it is worth knowing exactly how good it is.

The floor, and how slowly averaging lowers it. The noise remaining after averaging for a given time, for three source resistances at room temperature, over nine decades of averaging time — both axes logarithmic. Averaging for a time τ is equivalent to narrowing the bandwidth to about one over twice τ, so the noise falls as the square root of the time, which the figure measures off its own curves as an exponent of exactly minus a half. That is a discouraging law: a hundredfold improvement costs ten thousand times as long. The dashed curve adds a baseline that itself wanders as the square root of the time, at three nanovolts per root second, and their sum has a least value — 7.4 nanovolts at 3.03 seconds for the ten-kilohm source. Past that, averaging makes the measurement worse. Every low-level measurement lives inside that shape, and the reason the floor is where it is rather than somewhere else is half a kT per quadratic term in a piece of wire.
Fig. 5 The noise remaining after averaging, for three source resistances, over nine decades of averaging time. Averaging for a time τ is equivalent to a bandwidth of about one over twice τ, so the noise falls as the square root of the time — an exponent measured off the drawn curves as exactly minus a half. The dashed curve adds a baseline that itself wanders, and the sum has a least value past which averaging makes the measurement worse.

A square root is the least satisfying improvement available: always there, never enough. A hundredfold reduction in noise costs ten thousand times as long, so a measurement that takes a second to reach a microvolt takes three hours to reach ten nanovolts and three years to reach one.

And it only works against noise that is white. Every real instrument has a baseline that wanders — a temperature drifting, a contact potential changing, an amplifier’s offset moving — and such a drift grows with the averaging time rather than falling. Adding the two gives a curve with a minimum, and the minimum is where the measurement should stop: for the ten-kilohm source drawn here, at sixteen seconds and six and a half nanovolts, with everything after that worse than what was already available.

The existence of that optimum is the practical content of this essay, and it is why serious low-level measurement is not about averaging harder but about moving the signal away from the drift — chopping it, modulating it at a frequency where the baseline is quiet, and detecting it there. A lock-in amplifier is a machine for doing exactly that, and what it is escaping is not the thermal noise, which is flat and unavoidable, but the drift, which is neither.

The same relation, from the other side

There is a symmetry in the result that is worth naming because it is a theorem rather than a coincidence.

A resistor dissipates: put a current through it and energy is lost. A resistor fluctuates: leave it alone and a voltage appears across it. Those two facts are the same fact, and the expression connecting them contains the resistance in both — the thing that makes the resistor lossy is the thing that makes it noisy, in exactly the proportion that keeps it in equilibrium.

That is the fluctuation–dissipation relation, and Nyquist’s result is its first instance. Its other instances are the same statement in other apparatus: a pollen grain’s wandering tied to the viscosity of the water, a pendulum’s Brownian motion tied to its damping, an energy fluctuating by an amount tied to a heat capacity. In each case a response coefficient and a fluctuation are two readings of one quantity.

The practical face of it is a rule with no exceptions: anything that can absorb energy at a frequency will emit noise at that frequency. A lossy capacitor is noisy; a lossless one is not. A mirror that absorbs a part in a million adds a part in a million of a thermal field to the beam. A mechanical mount with internal friction shakes. Designing something quiet is the same problem as designing something lossless, which is why gravitational-wave detectors hang their mirrors on fused silica fibres — a material chosen for having almost no internal damping, because damping is noise. The same reasoning explains why a mode with an extremely sharp response is quiet everywhere except at its own frequency, where it is very noisy indeed — the dissipation is concentrated there and so is the fluctuation.

The other three noises, and why they are not this one

It is worth setting the thermal floor beside the noises it is usually confused with, because the distinction is what decides which design move helps.

Shot noise comes from charge being discrete: a current II carries a fluctuation 2qIΔf\sqrt{2qI\Delta f}, because the arrivals are a Poisson process rather than a flow. It has no temperature in it, it vanishes when the current does, and it is the same statistics a radioactive sample’s counting has. Against it, the move is more signal rather than less noise: the ratio improves as the square root of the current.

Flicker noise rises as one over the frequency, depends on how a device was made, and has no agreed general theory. It is what makes the drift in the averaging figure, and the move against it is to measure somewhere else in frequency.

Quantisation noise is the price of reading a number rather than a voltage, and it is set by the least significant bit rather than by any physics. It is the only one of the four that goes away by spending money.

The thermal floor is the one that is a theorem. The other three are properties of particular devices, particular currents and particular converters; 4kBTRΔf4k_BTR\Delta f is a property of being at a temperature, and a component that had none of it would be a component in which a count of accessible states had somehow stopped being a count.

Equilibrium, one temperature, and a resistance that is real

The derivation is for a resistor in equilibrium. A device carrying current, amplifying, or held at two temperatures is not in equilibrium and has noise that no equilibrium argument gives: shot noise from the discreteness of charge, flicker noise from slow fluctuations in the device, and generation–recombination noise in semiconductors. A transistor’s noise is dominated by the first two and the thermal part is a floor beneath them.

The resistance is taken as real and frequency-independent. A real component has inductance and capacitance, so its impedance has a phase and its noise is the real part of that impedance — which can be very different from the resistance measured with a meter, and which is why the noise of a resonant circuit peaks at its resonance.

One temperature is assumed throughout. A resistor in a system with parts at different temperatures delivers noise at an effective temperature that is a weighted average, and engineering an amplifier’s noise temperature below its physical temperature is possible and routine, by arranging that the part of it which is lossy is not the part that is warm.

The quadratic terms are assumed independent. Equipartition gives half a kT to each quadratic term of a sum of independent terms, and the modes of a line are independent because they are the normal modes of a linear system. A nonlinear component in the circuit couples them, and then the accounting is the one a chain of masses needs rather than this one.

And the mode counting treats the line as one-dimensional and lossless. A real cable attenuates, which means it is itself a resistor distributed along its length, and a long cable’s noise is its own rather than its terminations’. The derivation survives because a lossy line in equilibrium delivers exactly the noise its loss requires, which is the fluctuation–dissipation relation applied to the cable.

Modes counted in a system with no discrete anything in it

The mode-count figure draws a convergence and cannot show what the modes are. A standing wave on a transmission line is a real oscillation of real charges — electrons moving a fraction of an atomic diameter, at every frequency at once, uncorrelated — and the picture of a cavity with a countable set of modes in it is a bookkeeping device for a system with no discrete anything in it.

The spectrum figure draws an average energy per mode and cannot show the distribution around it. The noise voltage at any instant is a Gaussian random variable, and everything about how a measurement behaves — how often a threshold is crossed, how a peak detector responds, what a short average looks like — depends on that distribution rather than on the mean it is drawn from.

And the averaging figure draws a smooth improvement, which is what happens to the expected noise. A particular measurement averaged for a particular sixteen seconds gets whatever it gets, and the spread of those outcomes falls as the same square root. Nothing in the curve says how confident one run should make anybody, which is the question a real measurement is about.

Still open: what a measurement can do about a fluctuation it cannot remove

The thermal floor is set by a temperature and a dissipation, and both are properties of the apparatus rather than of what is being measured. That leaves an obvious question: is it a floor on the measurement or only on this way of making it?

Mostly the latter, and the exceptions are the interesting part. A measurement can beat the thermal floor by correlating two channels, so that the uncorrelated noise averages away and a correlated signal does not — cross-correlation spectroscopy does this routinely and recovers signals well below either channel’s own noise. A measurement can be made at a frequency where the apparatus has almost no dissipation, which is what a high-quality-factor mechanical resonator is for. And a measurement can be made on a system prepared in a state whose fluctuations are not thermal at all, which is what squeezed light is and what gravitational-wave detectors now use.

What remains underneath is not thermal. When the temperature is low enough, the fluctuation left is the zero-point motion of the modes themselves, which does not go away and which is the same hf/2hf/2 that the quantum expression above sits on top of. Whether that floor can be evaded — by measuring a quantity that does not disturb its conjugate, which is the programme of quantum non-demolition measurement — is a question with working experiments and no settled general answer.

The habit worth carrying away is the one the derivation is: a theorem about quadratic terms is a theorem about anything with quadratic terms. Equipartition was invented for gases and has nothing to do with gases. Every time it has been pointed at a new system — a solid, a chain, a suspended particle, a wire — it has produced a number that was measurable and had not been predicted, and the only work was identifying what the quadratic terms were.

Part 6 of 7

This essay is one argument about Equipartition. The others:

The condition three modes never quite satisfy. By how much three modes of a thirty-two mass chain fail to be in resonance — the sum of two mode frequencies minus the frequency of their sum, on a logarithmic scale, against the second of the two modes for four choices of the first. The leading nonlinear term couples modes in threes and the exchange accumulates only where this quantity is zero. It never is: the chain's dispersion is a sine, a sine is concave, and the sum of two of its values always exceeds the value at their sum. The smallest mismatch anywhere on the chain is at the two lowest modes and equals 2.156e-4 — which the scan finds and which is the cube of pi over four times the cube of one more than the mode count, checked here on chains from eight masses to two hundred and fifty-six. That closed form is the whole of why this is a finite-chain problem: the mismatch falls as the cube of the length, so a long enough chain is arbitrarily close to resonant and the continuum limit is exactly resonant, which is where the solitary waves come from. The condition three modes never meet Part 5 — Whether two modes of a chain can hand energy to a third is arithmetic on the dispersion relation, and for a chain of masses the answer is never: a sine is concave, so the sum of two frequencies always exceeds the frequency of their sum. The smallest shortfall anywhere on a chain of N masses is π³/4(N+1)³ — never zero, and never far from it — and a shortfall turns a transfer into a beat. Twice the kinetic energy, and what it equals. Twice the time-averaged kinetic energy of a bound orbit, divided by its time-averaged potential energy, against the power with which that potential depends on separation. Each point is a measurement: an eccentric orbit integrated for more than a hundred radial periods, with the two averages accumulated along it, and the radius checked to vary by at least a fifth so that the orbit is not trivially circular. The line is the exponent itself, and the points miss it by at most 6.6e-4. Two cases carry everything. At n = 2, a harmonic well, the two energies are equal — which is the ordinary equipartition statement, half a kT to the kinetic term and half a kT to the potential one. At n = −1, which is gravity and the Coulomb force, twice the kinetic energy equals minus the potential energy, so the total energy of a bound system is minus its kinetic energy. Nothing about temperature entered, and nothing about equilibrium: the relation holds for one orbit averaged over time as well as for a crowd averaged over members. Weighing what cannot be put on a scale Part 7 — Summed over every coordinate of a bound system, equipartition stops being a statement about temperature and becomes a relation between two averages: twice the kinetic energy equals n times the potential energy for a potential going as the nth power. For gravity that fixes a bound system's total energy from how fast its parts move — so a Doppler shift and an angular size return a mass, and for the Coma cluster the mass they return is fifty times the mass that shines.

Where this goes next

The staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant. Half a kT for every way of moving A heat capacity ought to be a count. Every quadratic term in a system's energy carries half a kT of it, so warming a gas is a matter of enumerating the ways its molecules can move — and the fact that the count comes out wrong for hydrogen is how a thermometer measured Planck's constant. A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 5, 12, 20 times kT are factors of 10^-2.2, 10^-5.2, 10^-8.7. At 295 K, kT is 25.4 meV, so a barrier of 0.4 eV is 15.7 kT and a factor of 1.5e-7. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens. The temperature a molecule does not have Temperature fixes a system's average energy and nothing more. The actual energy wanders, by an amount tied to the heat capacity, and the relative size of the wandering falls as one over the square root of the number of degrees of freedom — so a mole has a temperature and a molecule does not. Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run. The jiggle that proved atoms A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

BlackbodyDegrees of freedomEquipartitionFluctuation dissipationJohnson noiseMeasurementNoise floorStatistical mechanicsTemperatureThermal equilibrium