Illustrated Physics
physics, drawn
Thermodynamics

The noise a high Q moves out of the way

Anything held by a spring at a temperature jiggles, and equipartition says exactly how much: the mean square of its displacement is kT divided by the spring's stiffness, whatever the spring is made of and however it is damped. That looks like a floor no measurement can go below. It is a floor on the total, and the total can be moved. An oscillator with little damping puts almost all of its thermal motion into a narrow peak at its own frequency and leaves every other frequency quiet — which is why the mirrors of gravitational-wave detectors hang from fibres of glass that ring for minutes.
15 min read 5 figures What stays the sameMade into an instrument

Assumes: Half a kT for every way of moving · Half a kT in a piece of wire

Half a kT for every way of moving gave equipartition as a rule for storing heat: every quadratic term in a system’s energy holds, on average, half of kTkT. Half a kT in a piece of wire turned the rule into a noise: a resistor at a temperature produces a fluctuating voltage whose power is fixed by that same half kTkT, and no design of resistor can avoid it. And weighing what cannot be put on a scale summed the rule over a whole bound system and made it an instrument. The question left, and the one this essay takes up, is what a noise fixed by equipartition actually constrains.

The answer turns out to be less than it seems. Equipartition fixes how much a spring-mounted mass moves on average — its mean-square displacement is kT/kkT/k, where kk is the spring’s stiffness. It says nothing about how fast that motion happens, how it is spread over frequency, or how much of it falls in any particular band. Those are decided by the damping, and a measurement that listens only to some frequencies can have far less thermal noise than the equipartition total suggests. The whole art of the most sensitive mechanical measurements ever made rests on this.

The same area, three shapes

The same thermal motion, spread three ways. How the thermal jiggling of a spring-mounted mass is distributed over frequency, for quality factors 3, 30, 300, with frequency in units of the resonance and the spectrum in units that make the area under each curve its mean-square displacement. All three curves enclose exactly the same area, kT/k — 1.000, 1.000, 1.000 of it in the units drawn — because equipartition fixes the energy in the spring and not how fast it changes. What the damping chooses is where that motion goes: a high Q concentrates it in a peak 100 times taller than the low-Q one, and leaves the rest of the spectrum 100 times quieter at 10 times the resonance frequency.
Fig. 1 The thermal displacement spectrum of a spring-mounted mass at Q = 3, 30 and 300, frequency in units of the resonance, scaled so the area under each curve is its mean-square displacement. All three enclose exactly kT/k. The Q = 300 peak is 100 times taller than the Q = 3 one; at ten times the resonance frequency its spectrum is 100 times lower.

A mass on a spring in a gas or liquid is kicked continually by the molecules around it. The kicks are random, and the mass responds as any oscillator responds to a force: strongly near its resonant frequency, weakly far from it, with a peak whose width is set by the damping. The drawing shows the spectrum of the resulting motion — how much of the mean-square displacement lies at each frequency — for three different dampings, measured by the quality factor QQ: the number of radians of oscillation over which the energy falls by a factor of ee.

The three curves have very different shapes. At Q=3Q = 3 the spectrum is a broad hump. At Q=300Q = 300 it is a spike at the resonance, a hundred times taller, and away from the spike it is a hundred times lower. But the area under all three is the same. It is kT/kkT/k, exactly, and the drawing checks it by integrating each. That area is equipartition: the spring’s potential energy averages half kTkT whatever the damping, so the mean-square displacement is fixed.

The damping cannot change the total because the damping and the kicks are the same thing. A gas that damps an oscillator strongly is a gas whose molecules hit it hard and often; the same collisions that take energy away when the mass moves give it energy at random when it does not. Make the damping weaker and the random force weakens in exactly the proportion that keeps the energy at half kTkT. That is the fluctuation–dissipation relation, and the resistor’s noise is the electrical case of it: the same resistance that dissipates a current generates a noise voltage in proportion.

Calm and jittery, at one temperature

Two oscillators with the same temperature and the same average motion. The displacement of a spring-mounted mass kicked at random by the molecules around it, over 120 time units of 1/ω₀ (about nineteen cycles), for Q = 3 and 300, with the random kicks made stronger where the damping is stronger, as the fluctuation–dissipation relation requires. Dashed lines mark the root-mean-square displacement √(kT/k). The low-Q mass jitters, its motion changing direction at random every fraction of a cycle. The high-Q mass swings almost like a pendulum, a clean oscillation whose amplitude wanders slowly over hundreds of cycles. Averaged over a long run, both have the same mean square: 1.00 and 0.92 kT/k. An eye would call the second one calmer. A thermometer would call them identical.
Fig. 2 The displacement of a spring-mounted mass kicked at random by its surroundings, over about nineteen cycles, at Q = 3 and 300, with the kicks scaled to the damping as the fluctuation–dissipation relation requires. Dashed: the rms kT/k\sqrt{kT/k}. The low-Q mass jitters; the high-Q one swings almost like a pendulum with a slowly wandering amplitude. Over a long run their mean squares are 1.00 and 0.92 of kT/k.

The drawing makes the difference visible in time. Both oscillators are simulated with random kicks whose strength is set by their damping. The one with Q=3Q = 3 jitters: its motion changes direction at random every fraction of a cycle and looks like noise. The one with Q=300Q = 300 swings like a pendulum, almost a clean sinusoid, and its amplitude wanders slowly, growing for a hundred cycles and shrinking for another hundred, as the rare, weak kicks add or subtract a little each time.

Over a long run the two have the same mean square. Anyone watching them for a few cycles would call the second one calmer, because at any moment it is doing something regular. A thermometer — or equipartition — would call them identical. The difference is in their memory. The high-Q oscillator remembers its phase and amplitude for about QQ cycles, so its motion in any short stretch is predictable, and a measurement that knows what to subtract can remove it. The low-Q one forgets within a cycle, and its motion is new noise at every instant.

The motion added up from below

Where the thermal motion is, added up from low frequencies. The share of the total thermal mean-square displacement lying below each frequency, for Q = 3, 30, 300. Every curve ends at one — the whole of kT/k — but they get there differently. Below half the resonance frequency the low-Q oscillator already has 13 per cent of its motion and the high-Q one 0.1 per cent; the high-Q one gathers almost all of it in a step at the resonance, within a width of about f₀/Q. The step is equipartition's half kT, packed into a band 1/300 of the resonance frequency wide.
Fig. 3 The share of the total thermal mean-square displacement lying below each frequency, for Q = 3, 30 and 300. All end at one. Below half the resonance frequency the Q = 3 oscillator already has 13 per cent of its motion and the Q = 300 one 0.1 per cent; the high-Q one gathers almost everything in a step at the resonance about f0/Qf_0/Q wide.

Added up from low frequencies, the spectra show where the motion lives. The low-Q oscillator accumulates its motion gradually, with a good share of it at frequencies well below and above its resonance. The high-Q oscillator has almost none of its motion anywhere except in a narrow step at the resonance, of width about one QQ-th of the resonant frequency. Its half kTkT is all there, packed into a sliver of the spectrum.

That is the whole trick in one picture. A measurement made at frequencies below half the resonance sees thirteen per cent of the low-Q oscillator’s thermal motion and a tenth of a per cent of the high-Q one’s. The oscillators are at the same temperature and have the same total motion. One of them has moved its motion out of the way.

Why the tails fall as one over the square root of Q

The rate at which the thermal motion in a band away from the resonance falls with QQ follows from two facts already drawn. Far from its resonance an oscillator’s response to a force hardly depends on its damping: well above resonance the mass responds as a free mass, well below it as a bare spring. The random force driving it, on the other hand, has a strength set by the damping — by the fluctuation–dissipation relation its spectrum is proportional to the damping coefficient, and so to 1/Q1/Q. The motion in the band is the response times the force, so its mean square is proportional to 1/Q1/Q and its rms to 1/Q1/\sqrt{Q}.

At the resonance the same force meets a response that is QQ times larger, and the two effects of the damping cancel in the peak’s height times its width. That is the width that is a lifetime seen from the side of noise: a resonance whose width is one QQ-th of its frequency and whose peak is QQ times taller holds the same area, and the area is the half kTkT.

Why detector mirrors hang from glass

Thermal noise in a band away from the resonance falls as the Q rises. The root-mean-square thermal displacement that falls in a band of frequencies well away from an oscillator's resonance, in units of √(kT/k), against its quality factor on logarithmic axes: a band from 10 to 100 times the resonance, where gravitational-wave detectors listen above their mirrors' pendulum suspensions, and a band from a hundredth to a tenth of it. Both fall as one over the square root of Q: above the resonance from 1.5 × 10⁻² at Q = 1 to 1.5 × 10⁻⁴ at Q = 10⁴. The total thermal motion is the same at every Q. Raising the Q does not remove any of it; it moves it into a narrow peak at the resonance, out of the band where the measurement is made.
Fig. 4 The rms thermal displacement in a band away from resonance, in units of kT/k\sqrt{kT/k}, against Q: a band from 10 to 100 times the resonance and one from a hundredth to a tenth of it. Both fall as 1/Q1/\sqrt{Q}, above the resonance from 1.5 × 10⁻² at Q = 1 to 1.5 × 10⁻⁴ at Q = 10⁴.

A gravitational-wave detector measures the distance between mirrors kilometres apart to a precision far below the size of a proton, in a band of frequencies from about ten hertz to a few thousand. Each mirror is a forty-kilogram block of fused silica hanging as a pendulum, and the pendulum’s resonance is near one hertz — well below the band. Equipartition gives the pendulum a thermal motion of kT/k\sqrt{kT/k}, which for a forty-kilogram mass on a one-hertz pendulum is about 10−1210^{-12} metres, millions of times larger than the displacements the instrument has to hear.

What rescues the measurement is where that motion lies. The drawing shows the thermal motion falling in a band ten to a hundred times above the resonance, as a function of QQ, and it falls as 1/Q1/\sqrt{Q}. With a QQ of ten thousand the thermal motion in the band is a hundred times smaller than at Q=1Q = 1, and the pendulums in modern detectors have quality factors in the hundreds of millions. That is why the mirrors hang from fibres of fused silica welded to the mirrors themselves rather than from steel wires clamped to them: silica loses a smaller fraction of its energy per cycle than almost any other material, and a clamp is a place where rubbing surfaces lose energy. The same consideration applies to the mirrors’ own internal vibrations and to the thin reflective coatings on their faces, whose internal friction is at present one of the noises that limit the detectors in the middle of their band.

The logic is counter-intuitive enough to be worth stating plainly. The thermal noise in the band was not reduced by removing heat or by stiffening anything. It was reduced by making the oscillators ring longer — by reducing their damping — which concentrates their thermal motion into narrow peaks at their resonances, where the detector does not listen.

A resonance heard with nothing driving it

The thermal peak has a practical use that turns the argument around. An oscillator pushed at many frequencies at once answers most strongly at its own, the frequency that gets an answer, and the random force from the surroundings is exactly such a push — equal at every frequency, a flat spectrum of kicks. So the spectrum of an undriven oscillator’s thermal motion is a picture of its resonance: its peak is at the resonant frequency and its width is the frequency over QQ. An experimenter who wants to know a cantilever’s resonance and damping need not drive it at all. Recording its motion for a few seconds with nothing touching it, and computing the spectrum, gives both, measured by the molecules of the air.

This is how the resonances of nanomechanical devices far too small to drive cleanly are routinely characterised, and how the mirrors of a gravitational-wave detector have their suspension resonances identified: each mode shows up in the detector’s noise as a thermal peak at its frequency, and the width of the peak says how well it is isolated. The thermal noise that the design works to push out of the band is also the thermometer, the stopwatch and the ruler for everything in it.

A thermometer that measures a spring

How far a spring jiggles, from optical tweezers to a mirror. The root-mean-square thermal displacement √(kT/k) of anything held by a spring of stiffness k, at room temperature and at 4 K, on logarithmic axes. Marked at room temperature: optical tweezers, k ≈ 1.0 × 10⁻⁵ N/m, 2.0 × 10⁻⁸ m; AFM cantilever, k ≈ 1.0 × 10⁻¹ N/m, 2.0 × 10⁻¹⁰ m; a 40 kg mirror on a 1 Hz pendulum, k ≈ 1.6 × 10³ N/m, 1.6 × 10⁻¹² m; a quartz tuning-fork prong, k ≈ 1.8 × 10³ N/m, 1.5 × 10⁻¹² m. The formula knows nothing about what the spring is made of or how it is damped. Measuring a small spring's thermal jiggle and dividing kT by its mean square is how the stiffness of atomic-force-microscope cantilevers and optical traps is routinely calibrated.
Fig. 5 The rms thermal displacement kT/k\sqrt{kT/k} against spring stiffness at 300 K and 4 K. Marked at room temperature: optical tweezers (10⁻⁵ N/m, 20 nm), an atomic-force-microscope cantilever (0.1 N/m, 0.2 nm), a 40 kg mirror on a 1 Hz pendulum and a quartz tuning-fork prong (about 1.6–1.8 kN/m, 1.5 pm).

The same independence from damping that makes the total unremovable makes it useful. The mean square kT/kkT/k contains nothing but the temperature and the stiffness, so measuring one gives the other. The drawing spans the range of springs used in measurement. A microscopic bead held in the focus of a laser beam by optical tweezers is held by a spring of about 10−510^{-5} newtons per metre and jiggles by twenty nanometres, easily seen in a microscope. The cantilever of an atomic-force microscope, about 0.10.1 newtons per metre, jiggles by a fifth of a nanometre. A quartz tuning fork of the kind in watches, or a detector mirror on its pendulum, jiggles by about a picometre.

For the softer springs this is how stiffness is measured. The cantilever of an atomic-force microscope is calibrated before use by recording its thermal jiggle, fitting the resonance peak in its spectrum to separate the thermal motion from the instrument’s own noise, and dividing kTkT by the mean square. Optical traps are calibrated the same way, from the bead’s Brownian motion — the jiggle that proved atoms put to work as a ruler. In 1931 Eugen Kappler used a tiny mirror suspended on a quartz fibre in the same way in reverse: knowing the fibre’s stiffness and measuring the mirror’s thermal twisting, he found Boltzmann’s constant, and with it Avogadro’s number, to about one per cent. He also showed the two traces the essay’s second drawing shows — at high gas pressure a jittery twitching, at low pressure a smooth swinging — and found their mean squares the same.

The same rule for a capacitor

The electrical twin of the spring is a capacitor, and it has the same property in an even starker form. A capacitor connected to a resistor at a temperature holds a fluctuating voltage whose mean square is kT/CkT/C: the energy in the capacitor, 12CV2\tfrac12 CV^2, averages half kTkT. The resistor’s value does not appear. A large resistance makes a large noise voltage per unit of bandwidth but a narrow bandwidth, set by the product of resistance and capacitance; a small resistance, a small noise density spread over a wide band. The area under the spectrum is kT/CkT/C either way — about 64 microvolts for a picofarad at room temperature.

This fixed kT/CkT/C noise is a practical limit in every camera sensor. Each pixel is reset by connecting its small capacitance through a switch, and when the switch opens it freezes whatever voltage the thermal noise happened to leave, with an rms of kT/C\sqrt{kT/C}. No choice of switch reduces it; the cure is to measure the frozen voltage immediately after the reset and subtract it from the voltage after the exposure, which removes the reset noise because it is the same in both readings. It is the electrical version of a high-QQ oscillator’s predictability: noise that holds still long enough to be measured can be subtracted.

What equipartition does and does not promise

The distinction the drawings keep returning to is between a total and a distribution. Equipartition is a statement about an average over all frequencies, and it holds exactly for any classical oscillator in thermal equilibrium. It says nothing about any one frequency. The spectrum at a given frequency is set by the fluctuation–dissipation relation, which makes it proportional to how much the system dissipates at that frequency, and dissipation is something an engineer can choose.

It is the same distinction that matters for the energy that refuses to be shared among the modes of a nearly harmonic chain: equipartition describes where energy ends up if it has time to get there, and how long that takes is a separate question with a separate answer. A high-Q oscillator is one that takes a long time to exchange energy with its surroundings, and that slowness is exactly what keeps its thermal energy confined to its resonance.

Where the classical picture stops

Quantum oscillators. At high enough frequencies or low enough temperatures, ℏω\hbar\omega exceeds kTkT and the oscillator’s energy is no longer kTkT but approaches the zero-point value ℏω/2\hbar\omega/2; equipartition fails and the thermal noise becomes quantum noise, which can be pushed below its apparent floor in one quadrature by squeezing but not removed. For the pendulums and cantilevers here, at room temperature and kilohertz frequencies, the quantum regime is far away; for high-frequency mechanical resonators cooled to millikelvin, it has been reached.

Structural damping. The spectra drawn assume velocity damping, a force proportional to speed, as a gas or liquid gives. Much of the damping in solid materials is not of that kind: it is a fixed fraction of energy lost per cycle, independent of frequency. Such structural damping changes the shape of the spectrum away from resonance — it falls more steeply above resonance and rises towards low frequencies below it — but the total is still kT/kkT/k, and a high QQ still pushes the motion into the peak.

One mode. Real objects have many vibrational modes, each with half kTkT. A measurement between two resonances sees the tails of all of them, and designing an instrument means pushing every resonance out of the band or giving each a high QQ.

What the spectra leave out

They leave out the measurement itself. Reading the position of a mass disturbs it — a laser beam pushes on a mirror, an electrical sensor exerts a force on a cantilever — and that back-action is a noise of its own, which adds to the thermal motion and has its own spectrum. In the most sensitive detectors the quantum version of that back-action, the fluctuating pressure of the laser light, is comparable with the thermal noise at the bottom of the band. The spectra also leave out time-varying damping: a QQ that changes as a material ages or as a clamp settles moves the thermal noise around in the band, and tracking such changes is part of operating a detector.

Still open: the loss in the coatings

The noise that limits the current gravitational-wave detectors in the middle of their band is thermal motion of the mirrors’ reflective coatings, layers of amorphous oxides a few micrometres thick whose internal friction is far higher than that of the silica substrate beneath them. By the fluctuation–dissipation relation the coating’s loss sets its noise, and lowering it requires coatings with less internal friction — which means understanding, at the level of atoms rearranging in a glass, where amorphous materials lose mechanical energy. Crystalline coatings grown on semiconductors and amorphous coatings annealed or doped to reduce their loss are both being developed, and how far the loss can be reduced before the next detectors are built is not yet known.

The habit worth carrying away is to ask whether a limit is on a total or on a distribution. Equipartition fixes how much a system fluctuates in all, not where in frequency the fluctuation lies, and whatever sets the dissipation sets the distribution — so a sensor cannot be made to jiggle less than kT allows, but it can be made to jiggle only at the one frequency nobody is listening to.

Part 8 of 8

This essay is one argument about Equipartition. The others:

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. 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. 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. Where the instrument can hear. The response of an L-shaped interferometer to the plus polarisation, over the whole sky: azimuth across, cosine of the polar angle up, so that equal areas of the picture are equal areas of the sky. The plus pattern is largest directly overhead and underfoot and along the arms, and vanishes on four lines where a wave stretches both arms equally and the interferometer has nothing to compare. The average of the square over the whole sky is 0.2333, summed over a hundred and sixty thousand directions, and the two polarisations' averages add to 0.4000 — two fifths exactly, whatever the polarisation angle. Averaged over that angle as well, each polarisation contributes a fifth, and that fifth is what turns a strain sensitivity into a range: it is why a detector's quoted reach is substantially less than what it would manage for a source overhead. There is no direction in which the instrument is deaf to both polarisations at once, and none in which it is fully sensitive either. What the instrument actually hears A gravitational-wave detector is not a ruler laid against a stretching space. It is a clock comparison, its response falls to nothing at frequencies where the wave turns over while the light is still in the arm, and there are directions in the sky where it is deaf.

The objects named here

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

Brownian motionDampingEquipartitionFluctuation dissipationQuality factorResonanceSpectral densityThermal noise