The energy that refuses to be shared

Equipartition is one of the most useful results in physics and one of the least examined. Half a kT for every way of moving derives it and uses it: every quadratic degree of freedom of a system in equilibrium holds the same average energy, and that single sentence gives the heat capacity of a gas, the Dulong–Petit law for a solid, and the Brownian energy of anything small.

The derivation assumes equilibrium. Nothing in it says how a system gets there, or how long that takes, or whether it always does.

In 1953 Fermi, Pasta, Ulam and Tsingou put that question to one of the first computers. Take a chain of masses joined by springs, make the springs slightly nonlinear so the modes can exchange energy at all, put everything into the longest mode, and watch it spread. The expected result was a demonstration of how a system approaches equilibrium — a curve rising to a flat spectrum, useful as an illustration.

Fig. 1 The energy of the first five modes of a chain of thirty-two masses with springs a few per cent nonlinear, started with everything in mode one. The energy leaks into the next few modes, and after about a hundred and fifty periods it comes back — almost all of it. The dashed line is the equal share equipartition would give each of the thirty-two modes.

What the chain does instead

The energy leaves the first mode. That much is as expected: with linear springs the modes are exactly independent, so nothing at all would happen, and the point of the nonlinearity is to let them talk.

What follows is not. The energy goes into modes two, three and four, hardly touches anything above about mode six, and then reassembles. At a hundred and fifty-four periods of the longest mode, the first mode holds ninety-eight per cent of the energy again — and the cycle repeats.

Ninety-eight per cent is the number to hold on to. A system that had shared out its energy and happened to fluctuate back would do so once, by accident, and not on a schedule. A system that returns to within two per cent of its initial state repeatedly is not exploring the space available to it; it is moving on something much smaller.

The total energy is conserved to a part in ten thousand throughout, and that is checked against the chain’s actual Hamiltonian — including the cubic term the modes have no share in — rather than against the sum of mode energies, which is not conserved and would report the nonlinearity itself as an error of the same size as the effect.

What was actually run

The details of the calculation are worth a paragraph, because they are what makes the result a measurement rather than an anecdote.

Thirty-two masses, fixed at both ends, joined by springs whose force is the usual linear one plus a term proportional to the square of the extension. The nonlinear term is a few per cent of the linear one at the amplitudes the chain reaches — small enough that a physicist of the time would have called the chain harmonic, and large enough that the modes exchange energy visibly within a few dozen periods.

The integration here is velocity Verlet, which is symplectic: it does not conserve the energy exactly but its error does not accumulate, so a long run neither heats up nor cools down. That property is essential and not cosmetic. An ordinary integrator loses or gains energy steadily, and a chain slowly gaining energy would drive itself towards equipartition for reasons that have nothing to do with the physics — which is precisely the kind of artefact the original authors spent months eliminating.

The mode energies are read off by a sine transform of the positions and velocities at each sample, giving $E_k = \tfrac{1}{2}(\dot{A}_k^2 + \omega_k^2 A_k^2)$ for each of the thirty-two modes. Those are harmonic energies computed for a chain that is not harmonic, which is the right thing to do — the question being asked is where the energy would be if the chain were described in the variables that would be exactly right without the nonlinearity — but it means their sum is not the conserved quantity and must not be used as one.

How far from equipartition

The recurrence is dramatic and could be a peculiarity of the first few modes. It is not.

Fig. 2 The share of the energy each mode holds, averaged over the whole run, on a logarithmic scale. Equipartition is the flat line. The first four modes hold almost everything and the spectrum falls away by orders of magnitude above them; several modes hold less than a millionth of an equal share.

A chain with linear springs would show one bar and thirty-one zeros. A chain hit hard enough would show the flat line. This chain shows a third thing that no argument then available predicted: a spectrum confined to a corner of itself, decaying steeply, and stable for as long as anyone runs it.

The confinement is what makes the result a problem rather than a curiosity. Statistical mechanics is built on the assumption that a system with many degrees of freedom explores them, and this system has thirty-two and uses four.

What is happening in space

The mode picture says where the energy is. The chain itself says why.

Fig. 3 The displacement of each mass, at four times through the run. The chain begins as one smooth half-wave — that is what all the energy in mode one looks like — and the nonlinearity steepens it until the arch has broken into a train of narrow pulses running along it.

Zabusky and Kruskal noticed in 1965 that the continuum limit of this chain is an equation whose solutions are solitary waves — pulses that keep their shape as they travel, because the steepening that would break them is exactly balanced by the spreading that would flatten them. That balance is the subject of the pulse two failures keep alive, and it is the same mechanism here.

The property that matters for the recurrence is the one they had to check numerically before believing: two of these pulses pass through one another and come out unchanged. They are shifted along, and otherwise they are the pulses that went in. A collision between them scatters nothing into the rest of the spectrum.

So a chain that has broken into pulses has not lost track of where it started. The pulses run round, pass through each other, and periodically all line up again — and when they do, they add up to the smooth arch they came from. The energy in mode one is back because the shape is back.

Why the modes were the wrong variables

This is the deeper lesson and it is worth stating plainly.

The normal modes are the right variables for the linear chain, where they are exactly independent and each is a constant of the motion. Equipartition is a statement about a system exploring the space its constants of motion leave available. With thirty-two independent modes, that space is a point and nothing moves; with a nonlinearity, the modes stop being conserved and the space opens up.

But it does not open up all the way. A weakly nonlinear chain turns out to be close to a different system that is also exactly solvable — one with its own thirty-two conserved quantities, which are not the mode energies but complicated combinations of them. Being close to that system means having quantities that are nearly conserved, and nearly conserved quantities confine the motion nearly as effectively as exact ones.

That is the same argument as the last curve to go: a system perturbed away from an integrable one keeps most of its invariant surfaces, and only loses them as the perturbation grows. The chain is not exploring a thirty-two-dimensional space because it is trapped on something much smaller — and the trapping is not exact, which is why the story does not end here.

The energy density that changes the answer

If the invariants are only approximate, they can be broken. The question is what breaks them.

Fig. 4 A single number for how spread the energy is — the entropy of its distribution over modes, divided by its maximum, so that one is equipartition and zero is everything in one mode — followed against time for chains struck at four different amplitudes. The gentle ones rise a little and stop; the hardest one keeps climbing.

Nothing was changed between those runs but how hard the chain was hit. The springs have the same nonlinearity in every one. What differs is the strain, and therefore how large the nonlinear term is compared with the linear one at the amplitudes the chain actually reaches.

Fig. 5 The spread of energy at the end of a fixed run, against the energy per mass the chain was given, over nearly three decades. The curve is low and flat at small densities and climbs steadily above a knee. Below it, the sharing takes longer than the run; above it, it happens while the run is going on.

So equipartition is not false. It is slow, and the time it takes depends on the energy density in a way nothing in its derivation mentions. Below the threshold the sharing time grows faster than any power as the density falls, so a chain given a little energy will share it out eventually and eventually can be longer than any patience. Above the threshold the sharing happens on a timescale a run can see.

The original calculation ran at a density well below the knee, for a few thousand periods. It found what the chain does on that timescale, which is recur.

Two explanations, and what each is for

There are two accounts of why the chain behaves this way and they are usually presented as rivals. They are not; they answer different questions.

The solitary-wave account explains the recurrence. Pulses that pass through each other unchanged cannot lose the information about what they started as, so a configuration built from them comes back when they realign. It gives the period, roughly, from the pulses’ speeds and the length of the chain, and it is a statement about the continuum limit — which is to say about the low modes, since those are the ones the continuum approximation describes.

The near-integrability account explains the confinement. A weakly perturbed integrable system keeps most of its invariant tori, so the trajectory does not wander over the energy surface, and the fraction of the surface it can reach grows with the perturbation. It gives no period and it says nothing about pulses; what it gives is the threshold’s existence and the reason the sharing time grows so violently as the energy density falls.

Neither explanation alone predicts the position of the knee in the threshold figure, and getting that number out of either requires computing which combinations of modes can resonate at each order in the nonlinearity — the resonance conditions, which are arithmetic statements about the mode frequencies. The mechanism was understood decades before the timescale was, which is a common enough order of events to be worth expecting.

What this cost, and what it bought

The result was not published for years. It looked like a mistake — a bug in the code, a step size too large, an artefact of a machine with a few kilobytes of memory — and a great deal of effort went into eliminating those possibilities before the conclusion was accepted.

What it bought was two subjects. Solitons came out of the attempt to explain the recurrence, and are now a standard part of nonlinear wave theory and of optical fibre engineering. And the question of how a nonlinear system with many degrees of freedom actually reaches equilibrium — how long it takes, whether it always does, what the timescale depends on — became a research programme that is not finished.

The habit the result punishes is treating equipartition as a fact about a system rather than about a limit. It is a statement about the distribution a system settles into given enough time, and “enough” is a quantity that has to be computed and sometimes exceeds the age of the universe.

That failure mode is not confined to toy chains. A glass is a system that has not reached equilibrium on any timescale anyone will wait for, and calling its heat capacity an equipartition result is a claim about a timescale rather than about a state. Why heating a perfect spring changes nothing makes the complementary point from the other end: a purely harmonic solid has no mechanism to share energy at all, and every real solid’s thermal behaviour depends on the nonlinearity being there.

Where the same thing happens for real

A chain of thirty-two masses is a model, and the behaviour it shows is not confined to models.

A crystal is a chain of masses with slightly nonlinear springs, in three dimensions and with a great many more of them. Its thermal conductivity is a statement about how quickly energy put into some modes finds its way into others, and the fact that a perfect harmonic crystal has infinite thermal conductivity — nothing scatters, so a heat pulse crosses it at the speed of sound and never spreads — is the same observation as this chain’s linear limit doing nothing at all. Real thermal conductivity is a measurement of the anharmonicity, and the frequency a lattice cannot carry is where the mode structure it acts on comes from.

Low-dimensional systems are where the analogy stops being loose. Heat conduction along a one-dimensional chain does not obey Fourier’s law: the conductivity grows with the length of the chain rather than staying fixed, because the mechanisms that would randomise the energy are as weak as this figure suggests. That is measurable in carbon nanotubes and in polymer chains, and it is a direct descendant of this calculation.

The general statement is that a system’s approach to equilibrium is a property of its interactions and not of its size. A large system with weak coupling between its parts can be further from equilibrium, for longer, than a small one with strong coupling — which reverses the intuition that many degrees of freedom means fast thermalisation.

Where the model stops

The chain is one-dimensional and has fixed ends. Both matter. A two-dimensional or three-dimensional lattice has far more ways for modes to satisfy the resonance conditions that let them exchange energy, and reaches equipartition at much lower energy density. Almost everything striking about this problem is a property of one dimension.

The nonlinearity is quadratic. The quartic version of the same chain behaves differently in detail — no recurrence as clean, a different threshold — and the differences are not small. Which nonlinearity a real system has is not a decoration.

Thirty-two masses is not many. The threshold’s position depends on the number of masses, and how it depends on it is one of the questions the subject is still arguing about, because the two proposed scalings differ in whether the effect survives at all in a large system.

And the runs here are short. A hundred and twenty periods of the longest mode is enough to see the recurrence and the threshold and is far too short to settle what happens after a million. Everything said above about “eventually” is an extrapolation, and the extrapolation is what the disagreement is about.

The name, and who was left off it

The problem is usually called Fermi–Pasta–Ulam, and for fifty years it was called that in print. Mary Tsingou wrote the code.

That is not a footnote about credit alone; it is a fact about how the result was produced. The calculation was among the first physics problems run on a stored-program computer, and getting it to run at all — laying out the arithmetic, choosing the step size, arranging the output so that thirty-two mode energies could be read at all — was the larger part of the work and was not separable from the physics. The recurrence was seen in output that somebody had to design in order for it to be seeable.

The report was circulated in 1955 as a Los Alamos document with three names on it, Fermi having died before it was written. The fourth name was restored to common usage only after 2008. A calculation is an experiment, and this one had an experimenter.

What the pictures cannot show

The spectrum figure is a time average, and time averages hide the thing that makes this problem what it is: the distribution is not steady, it is cycling. A steady spectrum concentrated in four modes and a cycling one that spends its time in four modes look identical on that plot and mean quite different things.

The entropy figure compresses a thirty-two-dimensional distribution into one number, which is what makes it comparable across runs and what makes it blind. Two chains with the same spread can have their energy in completely different modes — one in the lowest four and one in the highest four — and the figure cannot tell them apart. It is the right summary for asking whether sharing has happened and the wrong one for asking what was shared.

Where the ladder goes next

The equipartition ladder began with half a kT for every way of moving and the heat capacities it predicts, went on to the share that is not half a kT where a degree of freedom is not quadratic, and to the temperature a molecule does not have, where the concept is asked of something too small to carry it. This rung asks the remaining question — whether a system that ought to equipartition actually does — and finds that the answer is a timescale rather than a yes.

The rung after it is the one where the timescale is computed rather than measured: the resonance conditions that decide which modes can exchange energy at each order in the nonlinearity, and the theory that turns those conditions into a rate. The habit worth carrying is the one this rung is built on: when a result is derived for equilibrium, ask what sets the time to reach it, because the derivation does not say and the answer is sometimes longer than the experiment.

The energy that refuses to be shared