How many ways there are to vibrate

Only some notes fit on a string, and which ones is a question about that particular string. This essay is about a different question — how many fit below a given frequency — which turns out to have an answer that barely depends on the object at all.

Counting instead of solving #

The modes of a rectangle are known exactly: with the edges held, the wavenumber of the mode with $m$ half-waves across and $n$ along is $k = \pi\sqrt{m^2/a^2 + n^2/b^2}$. Every mode is a lattice point in the quarter-plane of positive integers, and asking how many modes lie below $k$ is asking how many lattice points lie inside a quarter-ellipse.

That question has an obvious answer and a subtlety. The obvious answer is the area of the quarter-ellipse, which works out to $Ak^2/4\pi$ with $A$ the area of the membrane. The subtlety is the boundary: lattice points on the axes are excluded, because a mode with $m=0$ is a membrane at rest, and the correction is proportional to the perimeter of the region.

A string is the one-dimensional case and the one where the counting is trivial: its modes are evenly spaced in frequency, so the number below any frequency is that frequency divided by the spacing, and there is nothing to estimate. Nothing about that generalises to two dimensions except the idea of counting itself — which turns out to be the part worth keeping.

Weyl proved in 1911 that the leading term is universal — that for any reasonable region in the plane the number of modes below $k$ is $Ak^2/4\pi$ plus something smaller, whatever the shape — and the perimeter term was conjectured at the same time and proved much later. Written out:

$$N(k) = \frac{Ak^2}{4\pi} - \frac{Lk}{4\pi} + \cdots$$

with $L$ the length of the boundary. The minus sign is the statement that a fixed edge costs modes: the boundary is where the wave must vanish, and the more boundary a region has for its area, the fewer ways there are to fit a given wavelength into it.

Three shapes, one area #

The claim that the count depends on the shape only through two numbers is testable, and the test does not need any approximation: there are shapes whose spectra are exactly known.

A rectangle is one. A right isoceles triangle is another, because it is half of a square and its modes are the square’s modes that vanish on the diagonal — which are the ones with $m > n$. So a square, an oblong and a triangle can all be given exactly the same area and their modes counted exactly, with no numerical eigenvalue solver anywhere.

The three perimeters are 4, 4.83 and 5 for the same area, and those are exactly what separate the three lines. Subtract the perimeter term as well and the three collapse onto one another.

So a spectrum reports two things about a region without being asked: its area, from how fast the count grows, and its perimeter, from how far the count lags behind. Both are read off a list of frequencies by somebody who has never seen the shape. That is a striking amount of geometry to recover from a sequence of numbers — and it is also the whole of what can be recovered this way, which is the other half of the story.

Can one hear the shape of a drum? Kac asked it in that form in 1966. The answer is no: Gordon, Webb and Wolpert built in 1992 a pair of differently shaped regions with identical spectra. They have the same area and the same perimeter, necessarily, because the count says so — and they are not the same shape.

What the count is for #

Counting modes would be a curiosity if it were not the step on which several of the largest results in thermal physics turn.

That $\omega^2$ is the factor in front of the Planck distribution. A cavity’s radiation is its electromagnetic modes; the number of them per unit frequency is fixed by the volume and by the dimension and by nothing else; and multiplying that count by the average energy of a mode gives the spectrum.

The count is what makes a blackbody spectrum the shape it is. The rise on the short-wavelength side is the mode count winning and the collapse beyond the peak is the Boltzmann factor winning, and the curve that would not come down is what happens when only the first of those is included. The catastrophe was never a failure of the counting; the counting was right, and what was missing was the factor that switches the high modes off.

The classical argument gave every mode $kT$, which is the equipartition result and was not the mistake. The mistake was arithmetic that had nothing wrong with it: an $\omega^2$ count times a constant energy per mode integrates to infinity. The ultraviolet catastrophe is a statement about how many modes there are, and Planck’s constant enters as a bound on what the high-frequency ones can hold rather than on how many of them exist.

The same $\omega^2$ count applied to sound waves in a crystal instead of light waves in a cavity gives Debye’s model of a solid’s heat capacity, with its $T^3$ at low temperature and a cut-off at the shortest wavelength the lattice allows. That cut-off is the one physical difference between the two problems: a cavity can hold arbitrarily short waves and a lattice cannot, because below the atomic spacing there is nothing left to wave.

Debye’s calculation of a solid’s heat capacity is the identical argument with sound in place of light and a highest mode in place of an unbounded spectrum. The $T^3$ law that comes out of it is one of the most accurately confirmed results in low-temperature physics, and every step of it after “count the modes” is bookkeeping.

How the universal part is proved #

The reason the leading term cannot depend on the shape is worth seeing, because the argument is elementary and explains what “universal” is doing here.

Cut the region into small squares. Two comparison problems can then be built. In the first, every small square is given a fixed edge of its own, which is a more restrictive boundary condition than the region had — the wave is now forced to vanish on lines where it was previously free — and adding constraints can only raise frequencies, so this count is a lower bound. In the second, every internal edge is left free, which is less restrictive, and that count is an upper bound. Both are sums over squares of the count for a single square, which is known exactly.

As the squares are made smaller the two bounds close on each other, and what survives is the area divided by the area of one square, times the count for one square. Nothing about the region’s outline survives that limit except how much area it enclosed. The boundary contributes only through the squares it cuts, whose number is proportional to the perimeter — which is the second term, arriving as the error of the first.

That is Weyl’s argument in outline, and it is the reason the law’s leading term holds for regions with corners, with holes, and with boundaries too rough to have a tangent anywhere.

The frequency at which a room stops having modes #

The staircase and the smooth curve describe the same object and are useful at opposite ends, and the frequency where the description changes over can be worked out.

Modes have widths, because a real room absorbs, and the width in frequency is set by how fast the sound decays. Below the frequency at which neighbouring modes start to overlap, a listener hears individual resonances and moving a metre changes the level of a tone by a great deal; above it, the modes are a continuum and the count is what matters. That crossover — the Schroeder frequency — is around 200 Hz in a domestic room and around 20 Hz in a concert hall, because the mode density grows as the volume.

It is why the same building acoustics that works well for a hall says nothing useful about a control room, why bass is the difficult part of every small listening space, and why the modes of a box are the right language at low frequency and a hopeless one at high. The same crossover appears in every system with a spectrum: individual states at the bottom, a density of states above.

The density of states #

The count differentiated is the density of states — how many modes there are per unit frequency, or per unit energy — and in that form it appears in nearly every calculation involving a large number of quantum levels.

The same law turns up again under another name. The allowed states of a particle in a box are standing waves subject to the same boundary condition as a drum’s, so the counting law for them is Weyl’s law with different symbols — and the electron count in a metal is that law evaluated at the Fermi level.

A metal’s electrons fill states from the bottom up, and the energy at which they run out — the Fermi energy — is set by requiring the count of states below it to equal the number of electrons. That is Weyl’s law used as an equation to be solved rather than as a description.

Which energies exist at all is the mode count; how many of them are filled is the statistics. The two are separate questions, answered separately and then multiplied, and keeping them apart is what makes a metal’s electronic heat capacity tractable: the density of states comes from the geometry of the box and the occupancy from the temperature, and neither knows about the other.

The same quantity decides reaction rates, emission rates and conductivities, always in the same shape: a rate is a matrix element times the number of states available to go to. Fermi’s golden rule is that sentence written as an equation, and the second factor is a count.

It is also what makes the exclusion principle produce a pressure: the electrons must occupy distinct states, the number of states below an energy is fixed by the volume, and squeezing the volume raises the energy of the topmost occupied one. Every step of that argument except the exclusion is mode counting.

Where the smooth estimate stops describing anything is a periodic medium. A band gap is a range of frequencies with no modes at all, so the counting function is flat across it, and Weyl’s smooth estimate is an average through the structure rather than a description of it. The gap a repeat opens is where the missing modes went: they are not destroyed but pushed to the band edges, and the total count is conserved — which is why the smooth law is still right about the total while being wrong everywhere in detail.

The difference between two infinities #

The most startling use of mode counting takes two counts that are each infinite and subtracts them, and gets a force that has been measured.

Every mode of the electromagnetic field has a zero-point energy of half a quantum, even with no photons in it. Summing that over all modes in any region gives infinity, and the infinity is usually discarded as an unmeasurable additive constant. What cannot be discarded is a difference: put two parallel conducting plates a distance apart and the modes between them are restricted — only wavelengths that fit are allowed — while the modes outside are not.

So the zero-point energy between the plates depends on their separation. Both the constrained and the free counts are infinite; their difference is finite, and differentiating it with respect to the separation gives a force. The plates attract, with a pressure

$$\frac{F}{A} = \frac{\pi^2\hbar c}{240\,d^4},$$

which contains no property of the plates at all — not their material, not their charge, not their mass. Only the geometry and two constants.

The fourth power makes it a short-range effect and a violent one. At a hundred nanometres the pressure is thirteen pascals, which is small and measurable; at ten nanometres it is over an atmosphere. That is why the force is irrelevant to anything at ordinary separations and is a dominant nuisance in micro-electromechanical devices, whose moving parts sit micrometres or less from their neighbours and occasionally snap together and stay.

Measuring it took until 1997 to do convincingly, and the difficulty is that everything else at that separation is also large — electrostatic patch potentials, residual gas, surface roughness. What makes the measurement possible is the exponent: nothing else in the vicinity falls as the fourth power of the separation, so sweeping the distance separates the Casimir term from its competitors by its shape rather than by its size.

The argument’s structure is worth carrying beyond its subject. Two divergent quantities whose difference is finite and observable is a pattern that appears throughout physics — in the self-energy of a charge, in the binding energy of a quantum field, in the running of a coupling — and the discipline in every case is the same: never compute either count, compute the difference, and check that the answer is independent of how the counting was cut off.

Changing the count changes the rate #

The density of states enters every rate, and the useful consequence is that a rate can be changed by changing the states rather than by touching the thing that is doing the emitting.

An excited atom emits a photon at a rate proportional to how many modes are available for the photon to go into at the transition’s frequency. In free space that number is the $\omega^2$ count of this essay, and the resulting lifetime is what atomic physics tabulates. Put the same atom inside a cavity and the count is a different number entirely.

Two things can happen. A cavity tuned to the transition concentrates the available modes into a narrow band around it, so the density of states at the atom’s frequency is much larger than in free space and the atom emits faster — by a factor that can be thousands. A cavity whose smallest dimension is below half the transition’s wavelength has no mode at that frequency at all, so the atom cannot emit, and its excited state’s lifetime is extended indefinitely.

Both have been done. The enhancement is used in light-emitting devices and in single-photon sources, where a fast, directional emission into one mode is exactly what is wanted; the suppression was demonstrated with atoms passed between closely spaced mirrors and with atoms in a waveguide below cutoff.

What makes it worth putting in this essay is the reassignment of blame. A spontaneous emission rate looks like a property of an atom, and it is not: it is a property of an atom and of the modes surrounding it, and the second factor is the mode count. An excited state has no lifetime of its own, only a lifetime given what it is allowed to decay into.

The same reasoning applies to a photonic crystal, whose band gap is a range of frequencies with no modes at all. An atom inside one, emitting into the gap, is in the strongest form of the suppressed case — and the prospect of building materials in which chosen transitions simply cannot happen is what the field was started for.

Where the smooth count is not enough #

Weyl’s law is asymptotic, and there are two quite different ways of caring about what it misses.

The first is that the staircase is a staircase. Between one mode and the next the count does not move at all, and near the bottom of the spectrum the smooth curve is a poor description of anything: a room at 40 Hz has a handful of modes and the estimate is useless, which is why small-room acoustics is a different subject from concert-hall acoustics. The crossover is the frequency at which the modes overlap, and it is where the language changes from listing modes to counting them.

The second is that the fluctuation of the true count about the smooth one is itself a subject. Its statistics depend on whether the corresponding classical motion is regular or chaotic — a rectangle and a stadium have quite different fluctuations with identical Weyl terms — and that observation is the foundation of quantum chaos. Order out of the random has a counterpart here: the smooth part of a spectrum knows the geometry, and the fluctuating part knows the dynamics.

The habit generalises past modes. Counting configurations rather than frequencies is the same manoeuvre thermal physics uses everywhere: a quantity hopeless to enumerate one at a time is easy to count in bulk, and the bulk count is what the physics actually needs. Weyl’s law is that habit applied to a wave equation.

The count that is not of modes #

One more use is worth naming because it looks like a different subject and is the same arithmetic.

The entropy of a gas is the logarithm of how many microscopic states it can be in, and counting those states means counting cells in a phase space of positions and momenta — which is the same lattice count in six dimensions rather than two. The volume factor that appears in every expression for the entropy of an ideal gas is that count, and the arbitrary cell size that classical statistical mechanics had to introduce to make it finite is Planck’s constant, arriving from the same direction it arrived from in the blackbody problem.

So the pattern is consistent across the subject: a quantity that is impossible to enumerate is easy to count in bulk, the count is a volume divided by a cell, and the physics is what multiplies it.

What the pictures cannot show #

Every count here is for a fixed edge. A free edge changes the sign of the perimeter term, and a mixed boundary gives something between. The area term is untouched, which is the sense in which it is universal, and the perimeter term is where the boundary condition shows up.

The shapes were chosen because their spectra are exact. A general region needs a numerical eigenvalue solver, and the agreement with Weyl’s law is then a test of two things at once. Using a square, an oblong and a triangle keeps the test on the law.

Nothing here is about degeneracy. The square has many modes sharing a frequency, and a small change of shape splits them without changing the count. That splitting is what a real drum’s timbre is sensitive to and the count is blind to.

And the third term is not a perimeter. After the area and the perimeter comes a constant that depends on the corners and on the curvature of the boundary — a disc and a square of the same area and perimeter differ in it — and beyond that the expansion stops being a series in any ordinary sense.

The ladder from here #

Later rungs on this anchor: the density of states as the object rather than the count; the same law in three dimensions with the surface term, which is what a cavity’s spectrum needs; the isospectral drums and what their construction actually does; and the spectral fluctuations, which are where the counting argument hands over to quantum chaos.

The neighbouring ladders are the modes of a string, where counting is trivial and the idea is visible, the drum’s own frequencies, which are what is being counted, and the blackbody spectrum, which is this count multiplied by an energy.