The condition three modes never meet

The chain that refuses to share its energy leaves one question standing. Equipartition is not false; it is slow, and how slow depends on the energy density in a way nothing in the derivation of equipartition mentions. That calculation measured the dependence and stopped there, because getting the timescale out of either of the two available explanations requires a calculation neither of them contains.

This essay is that calculation, and it turns out to begin with a piece of arithmetic that has no dynamics in it at all.

Which exchanges are allowed to accumulate #

The nonlinear term in the chain’s springs is quadratic in the strain, so when the chain is written in its normal modes that term couples them in threes. Mode $k_1$ and mode $k_2$ drive mode $k_3$, with $k_3$ fixed by the geometry to be $k_1 + k_2$ or $|k_1 - k_2|$.

A driven oscillator responds. Whether the response grows depends on whether the driving is at the oscillator’s own frequency. Drive mode $k_3$ at $\omega_{k_1} + \omega_{k_2}$, and if that happens to equal $\omega_{k_3}$ the response builds up cycle after cycle; if it does not, the response oscillates and returns what it took.

So the question of which modes can share energy is not a question about the coupling at all. It is a question about the frequencies, and specifically about whether

$$\omega_{k_1} + \omega_{k_2} = \omega_{k_1 + k_2}$$

can be satisfied. That is a resonance condition, it involves no amplitudes and no time, and for a chain it has an immediate answer.

The chain’s frequencies are $\omega_k = 2\sin\big(k\pi/2(N+1)\big)$ — the dispersion relation of a lattice, a sine, which is concave on the range of interest — and a concave function satisfies $f(a) + f(b) > f(a+b)$ for every positive $a$ and $b$. There is no exact three-wave resonance on a mass chain, ever, for any $N$.

And how badly it fails #

A statement that a condition is never satisfied is not yet useful. What matters is by how much, because the mismatch has to be compared with something.

The figure scans every pair on a thirty-two mass chain and finds the smallest shortfall at the two lowest modes, where it is $2.16\times10^{-4}$ of the chain’s frequency unit. That is not a numerical coincidence. Expanding the sine for small argument gives

$$\omega_1 + \omega_1 - \omega_2 = 4\sin\theta(1 - \cos\theta) \approx 2\theta^3, \qquad \theta = \frac{\pi}{2(N+1)}$$

so the least mismatch is $\pi^3/4(N+1)^3$, which the scan reproduces on chains from eight masses to two hundred and fifty-six.

That closed form is the whole reason this is a finite-chain problem. The mismatch falls as the cube of the chain’s length, so a long enough chain is arbitrarily close to resonant — and the continuum limit, where the dispersion becomes a straight line and every triple satisfies the condition exactly, is exactly resonant. Which is also where the solitary waves live: the pulse two failures keep alive is a continuum object, and the reason the continuum description works so well for the low modes of this chain is that those are the modes whose mismatch is smallest.

What a mismatch does instead #

The step from “no resonance” to “no sharing” is the one worth doing carefully, because the naive version of it is wrong: a mismatch does not stop two modes exchanging energy.

The two-oscillator problem is solvable and it says exactly what happens. The energy oscillates between them at a frequency $\sqrt{\Delta^2 + 4V^2}$, and the fraction that ever crosses is $1/(1 + (\Delta/2V)^2)$.

Two things in that expression decide everything above.

The transfer is capped rather than forbidden. At a mismatch of eight couplings, six per cent of the energy still crosses — and then returns, and crosses again, for ever. Nothing accumulates, which is the only sense in which the transfer “does not happen”.

And the cap depends on the mismatch relative to the coupling. That is the comparison the whole subject turns on, because the coupling in the chain is not fixed: the nonlinear term is proportional to the amplitude, so $V$ grows with how hard the chain is struck while $\Delta$ stays where the dispersion put it.

So the knee found empirically for the chain is the place where those two numbers cross. Below it the mismatch dominates, the exchanges beat and return, and the chain recurs. Above it the coupling dominates, the mismatch is irrelevant, and the energy spreads. The threshold is not a property of the nonlinearity or of the dispersion separately; it is the ratio of two quantities that live in different parts of the problem, and that is why no argument about either alone predicted it.

There is a further consequence in the beat frequency that is worth noticing because it inverts an expectation. A badly matched pair exchanges energy faster than a well matched one — the beat frequency grows with the detuning — while exchanging much less of it. Speed and completeness pull opposite ways, and an observer watching a rapid flicker of energy between two modes is watching a pair that is failing to share rather than succeeding.

A prediction the empirical curve could not make #

A mechanism earns its keep by predicting something the description it replaced did not. This one predicts something about the length of the chain.

The mismatch falls as $1/(N+1)^3$. So a longer chain, at the same energy density and watched for the same number of its own longest periods, is closer to resonance and should share its energy more readily. Nothing in the measured threshold curve says that — a threshold in energy density is silent about how many masses there are — and it is not obvious: a longer chain has more modes to fill and is slower, both of which pull the other way.

The controls are most of the work and are worth naming, because getting either wrong produces the result by accident. A fixed amplitude would give a longer chain more total energy, so the amplitude is chosen for each chain by bisection until the energy per mass matches. A fixed run time would give a longer chain fewer of its own oscillations, so the run is counted in periods of that chain’s longest mode. With both held, the only thing that differs between the six runs is how badly their resonance condition fails.

It rises, monotonically enough for a line through the points to have the sign the mechanism requires. A ninety-six mass chain at the same energy density shares its energy a good deal more readily than a sixteen-mass one, and the reason is arithmetic about sines rather than anything about how much energy is where.

That also supplies the missing half of the continuum story. In the limit of many masses the dispersion straightens, the mismatch vanishes, and every triple is resonant — so the continuum chain should thermalise instantly. It does not, because in the same limit it becomes the integrable equation whose solutions are solitary waves, and integrability forbids the sharing by a different route entirely. The two obstacles are on opposite sides of the same limit, which is why the problem is hard at every chain length and easy at neither end.

What is left when the leading process is dead #

A three-wave process that cannot accumulate does not end the matter. It contributes at second order in the nonlinearity to an effective four-wave process, that contributes to a six-wave process, and so on — each order weaker than the last by a factor of the nonlinearity, and each with its own resonance condition to satisfy.

That is the structure of the modern answer and it is worth stating plainly, including what is not settled.

The leading process is non-resonant, so the rate it would have set is replaced by a rate from a higher-order process. Each order down costs powers of the amplitude, so the thermalisation time rises steeply as the chain is struck more gently — which is what was measured as a threshold and is here a consequence of which resonance condition is the first to have solutions. Work since 2015 identifies six-wave interactions as the leading resonant process for this chain in the thermodynamic limit and predicts a thermalisation time falling as the eighth power of the energy density.

What is honest to say about that prediction is that it is a prediction. Exact resonances on a finite chain are a number-theoretic question about sines of rational multiples of π, and the answer depends on $N$ in a way that has nothing to do with physics: searching a chain of sixteen masses finds no six-wave resonance that changes any mode’s energy, a chain of twenty-four finds eight of them, and a chain of thirty-two finds none again. The thermodynamic limit smooths that out and a finite computation cannot.

The threshold, restated #

With the mechanism in hand, the curve measured empirically can be read as a comparison of two numbers rather than as a fact about the chain.

Below the knee, $\Delta$ beats $V$: the transfers are capped at a small fraction, they return what they take, and the chain recurs. Above it, $V$ beats $\Delta$: the mismatch is irrelevant, every channel is effectively resonant, and the energy spreads within a run.

So the threshold is not a property of the chain but a crossing between one number set by the dispersion and another set by how hard the chain was hit. That is why it moves when either is changed, why it depends on the chain’s length through the mismatch, and why no argument about the nonlinearity alone — and none about the near-integrability alone — located it.

The rate, measured over the range a computation reaches #

Which leaves the empirical question: over the range a run can cover, does a single rate law describe the chain?

A collapse is a specific claim and a limited one. If four runs at different amplitudes fall on one curve when time is rescaled by a power of the amplitude, then over the range collapsed they are the same process running at different speeds. If they do not, they are not.

They do, and the exponent is 1.90. That is a real measurement of the chain and it is not the eighth power above, and the difference is the reason the figure carries its own disclaimer.

These runs are all above the knee. The amplitudes are chosen large enough that the sharing happens within a run of forty thousand time units, which puts them in the regime where the coupling has already beaten the mismatch and the resonance conditions no longer gate anything. A single low exponent is what one would expect there. The weakly nonlinear regime — the one the 1955 calculation sat in, the one where the mismatch matters, the one the eighth power is about — is several decades of amplitude below the bottom curve, and reaching it would need a run longer by a factor no computation will do.

So the subject has a theory for a regime that cannot be simulated and a simulation for a regime the theory is not about, and the two have been argued over for seventy years for that reason. The honest statement is that the mechanism is understood and the rate is not measured where it matters.

Why this is a general shape rather than a peculiarity #

The structure — a leading process forbidden by a resonance condition, a rate set by the first process that is allowed, and a threshold where the coupling overwhelms the condition — recurs far outside this chain, and it is worth collecting because it makes the argument transferable.

Water waves. Deep-water gravity waves have a dispersion that forbids three-wave resonance for the same kind of reason, so the leading energy transfer among ocean waves is a four-wave process — and the spectral evolution of a sea is computed from the resonant four-wave manifold. The theory of it is the same theory.

Phonons in a crystal. The leading anharmonic process is three-phonon, and whether it is resonant depends on the dispersion: it is allowed in three dimensions, where there are enough directions for the momenta to add up, and heavily restricted in one. That is a large part of why a one-dimensional chain conducts heat so strangely and why thermal conductivity in one dimension does not converge.

Atoms and light. An atom absorbs a photon only near a transition frequency, and off resonance it borrows the energy and hands it straight back — which is what a refractive index is, and is the two-oscillator result with one of the oscillators being a mode of the field. A molecule re-radiating what passes it is doing exactly the capped, returned exchange the second figure draws.

And any driven oscillator at all. The statement that a response accumulates only on resonance is the same statement that a forced pendulum’s amplitude grows only when driven at its own frequency, which is the whole of what a driven oscillator does near its own frequency, with the driving supplied by another part of the same system rather than from outside. The quadratic potential that gives every mode a definite frequency in the first place is the leading behaviour of any minimum, and the nonlinear correction this whole essay is about is the next term in that expansion.

What an exact resonance and a finite chain assume #

The resonance condition is stated for exact resonances. Real systems have finite-width resonances: the modes have frequencies broadened by the nonlinearity itself, so a mismatch smaller than the width behaves as a resonance. The width grows with the amplitude, which is another way of stating the threshold, and computing it is the part of the theory that is hardest to make rigorous.

The counting of six-wave resonances is done on a finite periodic chain. The fixed-end chain the figures integrate has a different selection rule, involving reflections as well as sums, and its resonance structure is not the same. The mismatch figure uses the fixed-end frequencies, which is right for the chain being integrated; the six-wave counting quoted above is for the periodic case, which is what the literature uses.

The two-oscillator model has no back-reaction on the first mode’s frequency. A real nonlinear chain shifts every mode’s frequency in proportion to the energy in it, so the mismatch is itself amplitude-dependent — and the shift can either close the gap or widen it. That effect is the same order as the one being computed and the figure omits it.

And the collapse is over four runs and one decade of amplitude. A power law fitted over one decade is a weak measurement, the exponent moves by a tenth or two with the choice of runs, and nothing about the collapse says the same power holds a decade below.

A mismatch with nothing on the plot to compare it against #

The mismatch figure draws a frequency difference and cannot show the coupling it has to be compared with. A mismatch of $10^{-4}$ is enormous or negligible depending on a quantity that is nowhere on the plot, and every conclusion here is about the ratio rather than about either number.

The two-oscillator figure draws a clean periodic exchange, and the chain has thirty-two modes exchanging with each other simultaneously through hundreds of triples. What that looks like is not four beats of different depths but a quasi-periodic signal with no evident structure, out of which the recurrence emerges only because the low modes’ mismatches happen to be commensurate. Reducing it to two oscillators is what makes the mechanism visible and is exactly what throws away the reason the recurrence has a period.

And the collapse figure draws four curves lying on top of each other, which is evidence of a shared mechanism and is also the easiest thing in numerical physics to produce by accident. Two curves can be made to collapse by almost any rescaling; four is better; the honest statement is the factor of four by which the rescaling reduced the spread, which is quoted on the figure and is a modest number.

Still open: whether the recurrence and the rate are the same problem #

Two things have now been computed about this chain and it is not obvious that they are related.

The recurrence has a period, and the period comes from the low modes’ frequencies being nearly commensurate — the solitary-wave picture gets it from the pulses realigning. The thermalisation has a rate, and the rate comes from the first resonant process at whatever order it appears. One is a property of a handful of modes and the other is a property of the whole spectrum.

What is not known is how the first becomes the second. A chain that recurs for a hundred thousand periods and then does not has stopped doing something, and no calculation says what it stops doing or when. The candidates — a slow drift of the nearly conserved quantities, a gradual leak of energy into the high modes through the non-resonant channels, the breakdown of the invariant surfaces the last curve to go describes, or a chaotic region opening in the phase space — make different predictions about how the crossover depends on the number of masses, and the numerical evidence is compatible with more than one of them.

The habit worth carrying away is the separation this whole essay is built on. Whether two things interact and whether their interaction goes anywhere are different questions, and only the second is about the coupling. The first is about frequencies, it can be settled before any dynamics is computed, and settling it is usually far easier than computing anything. A great many systems that look strongly coupled do nothing at all, because every channel available to them is a beat.