Illustrated Physics
physics, drawn
Concept

Nonlinearity — where it appears

A response that is not proportional to what causes it, so that two inputs together do not give the sum of their separate outputs. It brings harmonics, hysteresis, thresholds and chaos, none of which a linear system can produce.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve.

The error that doubles on a schedule

Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

mechanics · Chaos Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time.

The map a dripping tap turns out to be

A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely.

mechanics · Chaos A drive at one frequency, and what comes back at three times it. The Fourier components of the steady motion of an oscillator driven at a single frequency ω = 1.35, for drive strengths of 0.02, 0.05, 0.1. The equation is a harmonic oscillator with a cubic term added, and the components are projected out of the integrated motion rather than assumed. A linear oscillator answers only in the first column. This one answers in the third as well, because x³ of a cosine contains a cosine of three times the angle. The third harmonic grows as the drive to the power 3.00 where the fundamental grows as the power 1.00, so it is negligible at small drive and not at large — which is why nonlinearity in an instrument is a specification rather than a yes or no.

The oscillator that answers at three times the question

Push a spring hard enough that the parabola stops being the whole story, and three things happen that a linear oscillator cannot do: it emits frequencies nobody supplied, its resonance leans over, and its amplitude at one drive frequency depends on where the drive has been.

mechanics · Harmonic approximation 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 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 How far the vacuum is from being a nonlinear medium. The size of the vacuum's departure from linearity, as a fraction, against the electric field it is subjected to — thirteen decades of field and twenty-eight of correction, both logarithmic. The scale is the Schwinger field, computed here from the electron's mass and the fundamental constants as 1.32e+18 volts per metre: the field at which a pair gains its own rest energy over a Compton wavelength, and therefore the field at which the vacuum stops being a passive backdrop. The marks are the strongest fields that exist. a laboratory magnet, 10 T is 2.3e-9 of it; a hydrogen atom's own field is 3.9e-7 of it; a 10²² W/cm² laser focus is 2.1e-4 of it; the Schwinger field is 1.0e+0 of it; a magnetar, 10¹¹ T is 2.3e+1 of it. So a laboratory is twenty-eight decades from making the effect large, and a magnetar's field is above the critical one — which is why the only places the vacuum's nonlinearity has been seen are the two where the fields are not human: the ultraperipheral collision of two heavy nuclei, and the surface of a neutron star.

The one medium that was supposed to add exactly

Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

waves · Superposition The four vortices a standing wave leaves behind. Streamlines of the steady flow that a standing sound wave sets up in a channel, over half an acoustic wavelength, with the horizontal axis in units of the wave's own phase and the vertical axis scaled to the channel. The sound itself is a back-and-forth motion that averages to nothing; this is what does not average to nothing. Four closed cells fill each wavelength, two above the centreline and two below, turning in opposite senses, with the fluid moving along the walls toward the velocity nodes and back along the centre. The boundary layer that generates all of it is 69 micrometres thick, which is 0.7 per cent of the channel and is thinner than the width of a line in this drawing. The cells are not in the layer; they fill the channel.

The drift a sound leaves behind

A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.

fluids · Viscosity The ring does not come back. A ring of freely floating masses at four phases of a passing gravitational wave and once after it has gone. The first four are the familiar picture: stretched one way, then the other, with the area unchanged. The fifth is the one the standard picture does not draw — the ring is permanently deformed, by 25 per cent of the largest deformation the wave itself produced, and nothing brings it back. The masses are not oscillating about a new centre; they are at rest, at new separations. Both the oscillation and the offset are exaggerated enormously: the real strain at 440 megaparsecs is 9.9e-22 and the real permanent offset is 2.5e-22, so the drawing magnifies both by about 2e+20. What is honest in the picture is the ratio between them.

The ring that does not come back

Every picture of a passing gravitational wave shows a ring of free masses stretched, squeezed and let go. The last frame is wrong. The ring ends a different shape — permanently, with the masses at rest at new separations — by about a fifth of the largest distortion the wave itself produced. What sources the offset is the energy the wave carried away, so the wave is remembering itself, and nobody has measured it.

astrophysics · Gravitational waves

Named alongside it

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

AnharmonicityChaosEquipartitionErgodicityIntegrabilityLyapunov exponentNormal modeNumerical experimentRelaxationResonanceSpectral entropyStability

All concepts