Illustrated Physics
physics, drawn
Concept

Ergodicity — where it appears

The assumption that a system's time average equals its average over all states of the same energy, which is what lets statistical mechanics predict anything. It is an assumption rather than a theorem, it fails for weakly nonlinear systems on any reachable timescale, and the failure is a statement about a time rather than about a state.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Entropy that is still there at absolute zero, counted and measured. Four substances whose entropy does not go to zero when they are cooled as far as anybody can cool them, with the entropy counted from the arrangements they froze into beside the entropy measured by integrating their heat capacities. Ice is the famous one: every oxygen has four hydrogen bonds and the rule is that two hydrogens sit near it and two far, which leaves six of the sixteen placements legal, and Pauling's count of the whole crystal collapses to R ln(3/2) = 3.371 J per mole per kelvin against a measured 3.41 — an agreement to one per cent from an argument on one line. Carbon monoxide and nitrous oxide are the easy cases, molecules that can lie either way round in the lattice and have too little to gain by choosing. The worst of the four is off by 25 per cent, which is the honest state of this subject: the counts are crude, they ignore the correlations between neighbouring choices, and they still land within sight of a calorimeter. What the figure is really about is that the third law has an escape clause and the escape clause is measurable. A perfect crystal has one arrangement and zero entropy; a crystal that ran out of time while it had many has the logarithm of however many it stopped at, permanently.

The entropy that is still there at zero

The third law says a perfect crystal has no entropy at absolute zero. Ice has 3.41 joules per kelvin per mole left over, and the number can be recovered from one line of counting — two hydrogens near each oxygen and two far, six legal arrangements out of sixteen, R ln(3/2). The law has an escape clause and the escape clause is measurable.

thermodynamics · Third law The energy that goes and comes back. A chain of 32 masses with springs a few per cent nonlinear, started with all its energy in its longest mode, with the energy of the first five modes followed against time in units of that mode's own period. The first mode gives up most of what it has — down to 9 per cent by 104 periods — and the energy appears in the second, third and fourth. Then it comes back: at 154 periods the first mode holds 98 per cent of the total again. Equipartition would put an equal share in every one of the thirty-two modes and leave it there. What happens instead is that a handful of modes trade with each other and return almost exactly to where they began, and go on doing so. The total energy is checked against its starting value throughout and holds to 9.2e-5, so nothing here is the integrator losing track of what it was given.

The energy that refuses to be shared

Put all the energy of a chain of masses into its longest mode and add a few per cent of nonlinearity, and equipartition says it should spread out among all thirty-two modes and stay there. It does not. It leaks into three or four neighbours and then comes back — almost exactly — and goes on doing so, and the calculation that found this was expected to be a demonstration that it would not happen.

thermodynamics · Equipartition The entropy a substance keeps depends on how fast it was cooled. The entropy of a supercooled liquid in excess of its crystal's, against temperature, for a substance melting at 305 K with an entropy of fusion of 43 joules per kelvin per mole and a liquid heat capacity exceeding the crystal's by 62. The equilibrium curve — the one the liquid follows while it can still relax — falls steadily and would reach the crystal's entropy at 152.4 kelvin. It never gets there, because the liquid falls out of equilibrium first, at a temperature that depends on how long it is given: 0.01 s/K freezes at 196.1 K with 15.61 left, 1 s/K freezes at 188.1 K with 13.03 left, 100 s/K freezes at 182.1 K with 11.03 left. Ice's residual entropy is a count and does not move; a glass's is whatever it happened to have when it stopped being able to change, and that is a property of the experiment.

The entropy that depends on how fast it was cooled

Ice's residual entropy is a count, and it comes out the same whoever measures it. A glass's does not. A glass keeps whatever entropy it happened to have when its own relaxation time crossed the experiment's, so cooling ten times more slowly leaves less behind — and extrapolating the equilibrium liquid below that point takes its entropy under the crystal's at a finite temperature, which cannot happen and does not, for a reason that is still argued about.

thermodynamics · Third law 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

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.

thermodynamics · Equipartition The noisy orbit a true orbit stays beside. A computation of the doubling map with an error of up to 0.001 added at every one of 400 steps, and two exact orbits compared with it. The exact orbit started from the computation's own first point parts from it at once, by more than 0.1 after 8 steps, because each error is doubled at every step after it. The shadow — the exact orbit built by running the map backwards from the computation's last point, where each error is halved instead — stays within 8.0 × 10⁻⁴ of the computation at every one of the 400 steps, closer than the noise added at any single step. Every point of the noisy computation is therefore a faithful picture of a real orbit of the map — just not of the one that started where the computation did.

The wrong orbit beside a right one

Every computer simulation of a chaotic system is wrong within a few dozen steps: rounding errors are stretched exponentially, and after fifty iterations of the simplest chaotic map two computations of the same equation disagree completely. Yet chaotic simulations are trusted everywhere, and for a reason that is a theorem rather than a hope. For a large class of chaotic systems the computed sequence, though it follows no true orbit from where it started, stays close to some other true orbit for as long as it runs.

mechanics · Chaos What one trajectory says, and what the crowd says. For the walk with α = 0.5: the time-averaged mean-square displacement of each of 12 single trajectories, each watched for 10000 time units — the average of the squared displacement over every window of a given lag along that one path — against the lag, on logarithmic axes (thin lines), and the mean-square displacement of an ensemble of 800 walkers at the same times (thick). The ensemble spreads as t^0.5. Every single trajectory's time average grows nearly in proportion to the lag — mean slope 0.82, where the ensemble's is 0.5 — as if it were ordinary diffusion, and the trajectories disagree with one another about the diffusion coefficient by more than a factor of ten. A time average along one path and an average over many paths, which for ordinary diffusion are the same thing, here give different laws. That is weak ergodicity breaking, and it is what a single-molecule tracking experiment has to be read against.

The diffusion that slows the longer it is watched

A random walker whose steps come at irregular times still diffuses normally, provided the waits have a finite average. Let the waits have no average — mostly short, occasionally enormous, as they are for a molecule that keeps getting stuck — and three things change at once. The spread grows more slowly than time. A diffusion coefficient measured by following one particle depends on how long it was followed. And following one particle for a long time no longer tells what many particles do: averages along a path and averages over a crowd give different laws. Tracking single molecules inside living cells runs straight into all three.

thermodynamics · Diffusion A peak that keeps growing while the light stays on. The momentum distribution of 20,000 atoms in a one-dimensional dark-state scheme after 100, 1,000 and 10,000 scattering times, as a density per unit of ħk, in bins a hundredth of ħk wide. There is no steady state. The peak at zero grows taller and narrower for as long as the light is on — its height reaches 29 per ħk by the last time — and the share of atoms within a tenth of a photon's momentum climbs from 55 to 87 to 96 per cent. The atoms are not pushed into the peak; they fall into it at random and the light stops touching them.

The atoms the light stops touching

Every photon an atom scatters kicks it by the same small amount in a random direction, so a gas that keeps scattering light cannot be colder than one kick. The way below is not a stronger friction but a weaker touch: arrange for the light to stop scattering off atoms that are nearly at rest. Then the coldest atoms are not cooled, only left alone, for times that have no average — and the gas never settles into a temperature at all, but piles into a peak that narrows for as long as the light stays on.

astrophysics · Radiation pressure

Named alongside it

The objects these essays reach for when they reach for this one.

RelaxationEntropyEquipartitionHeat capacityHeavy-tailed distributionIntegrabilityNonlinearityNormal modeNumerical experimentRandom walkResidual entropySpectral entropy

All concepts