Illustrated Physics
physics, drawn
Theme

The arrow of time

Nearly every law works equally well backwards. Almost nothing else does, and the gap between those facts is thermodynamics.
A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic. Mechanics

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

 Ways to arrange 10 coins. The number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it. Thermodynamics

Entropy is a count, and the arrow of time is arithmetic

Nothing in mechanics prefers a direction. Entropy is not a force pushing things toward disorder — it is the observation that some outcomes have vastly more ways of happening than others.

 A Carnot cycle on pressure–volume axes. Two isothermal steps joined by two adiabatic ones, forming a closed loop. The gas expands 9.6-fold in reaching the cold reservoir at 0.50 of the hot one, and the area enclosed is the net work done over one cycle. Thermodynamics

The ceiling on every engine, set before it was designed

There is a maximum efficiency no heat engine can exceed, and it depends on nothing but two temperatures. Not the fuel, not the working substance, not the cleverness of the engineer.

 A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand. Mechanics

The hill that gives it back, and the forces that do not

Potential energy turns a question about motion into a picture of a landscape. It works for gravity and springs, it fails for friction, and the difference between those two cases is the whole of what makes energy useful.

 Response against driving frequency. The steady-state amplitude of a driven oscillator against driving frequency, at three damping ratios. Lighter damping gives a taller and narrower peak, and the peak sits slightly below the natural frequency. Waves

The frequency that gets an answer, and the quarter cycle nobody mentions

Push an oscillator at its own frequency and the response grows enormously. The reason is not that the push is in step with the motion — at resonance it is a quarter cycle out, and that is precisely why it works.

 A spike, spreading. The solution of the diffusion equation at three times, with a seeded random walk histogrammed behind it. The area under every curve is the same because nothing is lost; only the width changes, and it grows as the square root of the time. Thermodynamics

The equation that only runs forwards, and the walk underneath it

A drop of ink spreads and never gathers. The equation describing it is one of the few in physics that is not reversible — and underneath it is nothing but a coin being tossed.

 A packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51. Waves

The packet that moves at another speed than its own crests

Watch a group of water waves and the individual crests run forward through it, rise in the middle, and vanish off the front. The group travels at half the speed of the crests, and both numbers are real.

 Heating 1 kg of water from -20°C to 130°C. Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey. Thermodynamics

The heat that changes no temperature, and where it actually goes

A kettle reaches a hundred degrees in a minute and takes five more to boil dry. The heat going in during those five minutes changes nothing a thermometer can see, and it is most of the energy in the whole process.

 Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that. Quantum

A nucleus with no clock

A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.

 One mass, two entropies. The entropy of a solar mass as ordinary gas, generously counted at ten Boltzmann constants per proton, against the entropy of a solar-mass horizon. The first is 1.19·10⁵⁸ k and the second 1.05·10⁷⁷ k — a factor of 8.83·10¹⁸. This is why a horizon had to be given an entropy: without one, dropping anything at all through it destroys entropy and the second law fails. Thermodynamics

The entropy that lives on a surface

Throw a cup of tea through a horizon and the entropy of the outside world falls. Either the second law is wrong or the horizon has an entropy of its own — and the only quantity available for it turns out to be its area, in units of a length made from gravity, quantum mechanics and the speed of light together.

 A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens. Thermodynamics

The exponential that decides everything

Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

 Ways to arrange 12 coins. The number of distinct arrangements giving each number of heads, for 12 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it. Thermodynamics

What a system actually minimises

A ball falls to the bottom of a bowl and a gas fills a room, and neither of those is the rule. A system in contact with a large reservoir minimises U − TS, and the minus sign is the reservoir's own entropy written in the system's variables — which is why a rubber band pulls harder when it is heated.

 How far apart the two paths can be. Fringe visibility against the difference between the two path lengths, for light at 550 nm with a bandwidth of 100 nm. Each curve is the modulus of the Fourier transform of its own line shape, summed over the spectrum here rather than taken from a standard result, and the three shapes have the same width at half height. The conventional coherence length λ²/Δλ is 3.02 µm for this light, and what the curves show is that the convention is a rounding of three genuinely different behaviours: a flat band halves at 1.83 µm, a Gaussian line halves at 1.33 µm, a Lorentzian line halves at 0.68 µm. The flat band comes back — a rectangle's transform rings — and the Lorentzian's tails keep a little visibility very much further out than its width suggests. Nothing here is about the apparatus: the fade is the source forgetting its own phase. Optics

How far a wave can remember

Split a beam, delay one half, and put them back together. The fringes are bright while the delay is short and fade as it grows, and the distance at which they die is fixed by nothing but the width of the source's spectral line. Watching them fade is reading the line shape.

 The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end. Waves

The packet that will not keep its shape

A group velocity is only the first thing a dispersion relation says. The second is that the packet spreads — at a rate fixed by its own bandwidth and by the curvature of ω(k) — so a short pulse comes apart quickly and a long one hardly at all. It is a trade quantum mechanics is usually given credit for, and classical waves make it too.

 The ceiling, inverted. How many joules of heat a perfect machine can move per joule of work, against the outside temperature, with the inside held at 21 °C. The upper curve is heating — T_h/(T_h − T_c), which is what the Carnot argument becomes when the cycle is run backwards — and the lower one is cooling the outside, T_c/(T_h − T_c). They differ by exactly one everywhere, to 1.8e-15 across the whole range as drawn, because the work put in is delivered as heat along with whatever was moved. The dashed line at one is a resistive heater, which is 100% efficient and is the worst option on the figure. At 7 °C and −7 °C the ideal coefficients are 21.0 and 10.5; a real machine reaching 25% of the ideal gets 5.3 and 2.6, which is still several times what burning the same energy would give. Thermodynamics

The engine that pays back more than it takes

Carnot's argument puts a ceiling on how much work a flow of heat can be made to do. Run the same cycle backwards and the ceiling inverts into a floor that is greater than one — so a machine can deliver three or four joules of heat for every joule it consumes, and a perfectly efficient electric heater is the worst way to warm a room.

 One equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2. Mechanics

The three ways of coming to rest

An oscillator with friction in it can swing and fade, arrive and stop, or crawl back so slowly it looks stuck. One equation and one number decide which. The number is not what most people would guess, and neither is the value that returns fastest.

 A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start. Electromagnetism

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

 The response of a machine with a 10% absorber bolted to it. The amplitude of the driven mass, in units of the deflection the same force would cause if applied slowly, against drive frequency in units of the machine's own natural frequency. Dashed: the machine alone, with its single resonance. Solid: the same machine with a second mass and spring attached, weighing 10 per cent of it and tuned to the same frequency. At that frequency the driven mass does not move at all — the amplitude is zero rather than small, and the absorber is moving 10.0 static deflections to make it so. What the device costs is the two new resonances it creates, at 0.854 and 1.171 times the original frequency, which are infinite in this undamped calculation and straddle the frequency the machine was protected at. Waves

The mass that makes another stand still

Bolt a small mass on a spring to a machine that is shaking itself apart, tune it to the frequency that is doing the damage, and the machine stops moving. Not moves less — stops, exactly, at that one frequency. The price is two new resonances either side of it, and the whole device is a bet that the drive stays where it was put.

 The viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make. Thermodynamics

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

 Where a drying drop loses its liquid. The rate at which liquid leaves the surface of a drying drop, against distance from the centre in units of the drop's radius, for 5 contact angles. The flux is not uniform, and it is not a property of the liquid: it is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the field around a charged lens and has the same answer — a power law in the distance from the rim, with an exponent that depends only on the contact angle. at 10° the exponent is 0.471, and the loss has doubled by 87.8 per cent of the way out, at 40° the exponent is 0.357, and the loss has doubled by 92.5 per cent of the way out, at 70° the exponent is 0.182, and the loss has doubled by 98.9 per cent of the way out, at 90° the exponent is 0.000 and the drop dries evenly everywhere, at 120° the exponent is -0.500 and the flux falls toward the rim. Below a right angle the flux diverges at the contact line; at exactly a right angle it is uniform; above it the edge is the slowest-drying part of the drop. Since a pinned edge must be resupplied from the interior, that sign decides which way the liquid inside the drop flows — and therefore whether everything suspended in it ends up in a ring at the rim or in a spot at the centre. Fluids

The ring the drop leaves behind

A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.

 The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals. Astrophysics

The size the light cannot blow away

Radiation pressure and gravity both fall as the inverse square of distance, so their ratio is a property of the grain and not of where it is. What follows is a band of sizes that get blown out — with a lower edge as well as an upper one — and a drag, on everything else, that is the same pressure read one order further in v/c.

 Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently. Electromagnetism

The magnet that has to fight its own field

A bar magnet's own poles put it in a reverse field, so the same material cut short and fat is weak and cut long and thin is strong. And a magnetised material does not have a magnetisation — it has a magnetisation and a history, which is why the word for what it does is the Greek for coming late.

 Solid and liquid are answers about a duration. The relaxation time of seven materials, on a logarithmic axis spanning 39 decades, against the length of one observation. A material behaves as a solid when its relaxation time is longer than the observation and as a liquid when it is shorter, so the vertical line is what decides which — and it is a property of the observer. At 1 s, 4 of these are solids. Move the line six decades to the right and pitch joins the liquids; move it far enough left and water is a glass, which is not a figure of speech but what a picosecond pulse measures. Fluids

The liquid that remembers

Pitch shatters like glass under a hammer and flows through a funnel over a decade. Neither behaviour is the true one. What decides which a substance shows is not the substance but the length of the observation, and the ratio between the two has a name and a number.

 How long an orbit has before the waves take it. The time a circular orbit has left before gravitational radiation brings it together, against its separation, for four pairs of masses. Both axes are logarithmic and every curve has the same measured slope, 4.00: the lifetime goes as the fourth power of the separation, so halving an orbit shortens its remaining life by a factor of sixteen. The Earth's orbit is 13 decades above the age of the universe and the neutron-star binary is below it, which is the whole difference between a system that is losing energy and a system that is going to merge. Astrophysics

The orbit that has to shrink

Two masses in orbit radiate gravitational waves and lose energy, so the orbit tightens, so they go faster and radiate harder. The runaway takes 10²³ years for the Earth and the Sun and eight minutes for the last thousand kilometres of a black-hole pair — and the same one-line formula gives both.

 The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does. Astrophysics

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

 What the return journey does to each of them. The rotation of the plane after a beam has gone through a rotator, been reflected, and come back, against the length of the rotator — divided by the one-way rotation, so the two answers are 0 and 2 and nothing else can happen. A naturally active medium is handed with respect to the beam: reverse the beam and the sense of the rotation reverses with it, and the second pass undoes the first exactly, at every length and every wavelength. A Faraday rotator is handed with respect to the field, which does not care which way the light is going, so the second pass adds to the first and the round trip is twice the single one: 1 mm of it gives 21.7° out and 43.4° back; 2 mm of it gives 43.4° out and 86.8° back; 4 mm of it gives 86.8° out and 173.6° back; 8 mm of it gives 173.6° out and 347.2° back. That is a violation of reciprocity, and it is only available because a magnetic field is odd under time reversal. Everything a passive optical component can do — a lens, a mirror, a waveplate, a piece of quartz — looks the same run backwards, and none of them can be made into a one-way street. This can. Optics

The rotation a return trip doubles

Quartz turns the plane of polarisation and so does glass in a magnetic field. The two look identical on the way through and are opposites on the way back — the crystal undoes its own rotation exactly, and the magnet adds to it. That difference is the whole of why a one-way street for light can be built at all, and why nothing passive will ever be one.

 A quarter of the time, an object is found without being touched. The outcomes of sending one photon into a balanced interferometer, with and without an opaque object in one arm, computed from the same amplitudes. With the arm clear, every photon leaves by the bright port and the dark port receives nothing. With the object in place, 50 per cent of photons are absorbed by it, 25 per cent reach the bright port and say nothing, and 25 per cent reach the dark port — which is impossible unless something is in the arm, and which happens with the object still sitting there unabsorbed and unlit. The photon that produced that click did not go through the blocked arm, because a photon that goes through a blocked arm is absorbed. So an object has been located by light that never met it. The price is that 50 per cent of attempts destroy the thing being looked for: an efficiency of 33 per cent, which the Zeno version of the apparatus takes as close to one as one likes. Quantum

The measurement that never touched it

A balanced interferometer sends every photon to one output and none at all to the other. Put an object in one arm and the empty port starts clicking — and a click there is caused by a photon that cannot have gone near the object, because a photon that goes near it is absorbed. The object has been found by light that never met it.

 The work one molecule and one bit are worth. The pressure of a gas of one molecule against its volume, at 300 K, in units of the volume it starts in. The shaded area is the work the molecule does pushing a partition out isothermally, and it is measured here by integrating the drawn curve rather than written down: expanding by 1.5× yields 0.4055 kT against ln 1.5 = 0.4055; expanding by 2× yields 0.6931 kT against ln 2 = 0.6931; expanding by 4× yields 1.3863 kT against ln 4 = 1.3863; expanding by 8× yields 2.0794 kT against ln 8 = 2.0794, agreeing to 2.0e-10. The doubling is the one that matters, because a partition inserted in the middle leaves the molecule on one side or the other, and knowing which is what lets the load be attached to the right face. That single expansion delivers kT·ln2 = 2.87 zeptojoules at 300 K. It looks like work extracted from one temperature, and it is — until the engine is asked to run again, which requires forgetting which side the molecule was on. Thermodynamics

The bit that has to be paid for

One molecule in a box, a partition, and the knowledge of which side it went — enough, between them, to extract work from a single reservoir, which the second law forbids. The engine is real and the arithmetic is right. What closes the loophole is that the cycle does not finish until the knowledge has been thrown away, and throwing away one bit costs exactly what the expansion delivered.

 The ceiling, the estimate, and three power stations. Two efficiencies against the ratio of the cold reservoir's temperature to the hot one. The upper curve is Carnot's 1 − Tc/Th, which is a ceiling on the work per unit of heat and is reached only by an engine that runs infinitely slowly, because a reversible heat flow needs a vanishing temperature difference to drive it and therefore infinite time. The lower curve is 1 − √(Tc/Th), the efficiency of an engine with finite thermal contact run for the most power rather than the most work. Three measured plants are marked: West Thurrock, coal runs at 36 per cent against a ceiling of 64 and a finite-time estimate of 40; CANDU, nuclear runs at 30 per cent against a ceiling of 48 and a finite-time estimate of 28; Larderello, geothermal runs at 16 per cent against a ceiling of 33 and a finite-time estimate of 18. Every one of them is closer to the lower curve — within 4.4 points at worst, against 16.5 at best from the ceiling. The second law is not what limits a working power station. What limits it is that somebody wants the electricity this year. Thermodynamics

The engine that has to finish

Carnot's ceiling is exact and it is reached only by an engine that takes for ever, because a reversible heat flow needs a vanishing temperature difference to drive it. Ask instead for the most power rather than the most work per joule of heat, and the answer is a different function of the same two temperatures — and three measured power stations sit on it rather than on the ceiling.

 What a thermal camera reads off a shiny surface. The temperature a camera calibrated for a black body reports, against the emissivity of the surface it is pointed at, for surfaces truly at 323 K, 373 K, 473 K in a room at 293 K. Every curve begins at the true temperature when the emissivity is one and ends at the room's temperature when it is zero, because a surface that emits nothing reflects everything and the camera is then looking at the room. polished aluminium at 0.05 reads 312 K; stainless steel, oxidised at 0.8 reads 451 K; matt black paint at 0.95 reads 468 K; human skin at 0.98 reads 471 K for a surface truly at 473 K. That is not a fault in the instrument. It follows from Kirchhoff's law — a poor emitter is a poor absorber and therefore a good reflector — so the shortfall in a shiny surface's own glow is made up almost exactly by whatever is reflected in it, and no measurement of the light leaving a surface can separate the two without knowing the emissivity in advance. Thermodynamics

The glow that says nothing about the surface

A thermal camera pointed at a saucepan of boiling water reads a hundred degrees. Pointed at a polished aluminium block at the same temperature it reads about twenty-five, and the instrument is working perfectly. What it is measuring is emissivity as much as temperature — and a surface's emissivity is forced to equal its absorptivity, at every wavelength and every angle, by an argument with no physics of matter in it at all.

 The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter. Astrophysics

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

 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. Thermodynamics

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.

 Cold matter, and the mass above which nothing holds it up. The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it. Astrophysics

The mass no cold matter can hold up

A white dwarf gets smaller as it gets heavier, which no ordinary object does. Follow that curve upward and the radius reaches zero at 1.46 solar masses — because once the electrons are relativistic the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball does not depend on its radius at all.

 Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else. Waves

The answer that cannot come first

A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives.

 The least energy that fresh water can cost. The work needed to take fresh water out of a feed of 1150 mol/m³ of dissolved particles — seawater — against how much of the feed is taken, in kilowatt-hours per cubic metre of product. The lower curve is the reversible minimum, in which the pressure is raised continuously as the remaining feed gets saltier; the upper one is a single stage held throughout at the pressure the final, saltiest concentrate needs. At vanishing recovery both tend to the feed's own osmotic pressure, 28.5 bar or 0.79 kWh/m³, which is the floor for the first drop and is checked here against the limit of the formula. Taking more of the feed costs more per unit taken, because what is left behind is saltier: at 10% recovery 0.83 kWh/m³ reversibly and 0.88 in one stage, 30% recovery 0.94 kWh/m³ reversibly and 1.13 in one stage, 50% recovery 1.10 kWh/m³ reversibly and 1.58 in one stage, 60% recovery 1.21 kWh/m³ reversibly and 1.98 in one stage, 75% recovery 1.46 kWh/m³ reversibly and 3.17 in one stage. The horizontal line is what a good seawater plant actually uses, 3 kWh/m³, so the second-law efficiency of the industry is about 37 per cent. None of this is about membranes. It is the free energy of mixing salt into water, read backwards, and no technology of any kind can go below the lower curve. Fluids

What it costs to take the salt out

Salt dissolves in water because mixing is overwhelmingly the more probable arrangement, and separating the two again means paying back what the mixing gave away. The bill can be computed before any apparatus is chosen — 0.79 kilowatt-hours for the first cubic metre from seawater — and it rises with every further cubic metre taken, because what is left behind is saltier than what was started with.

 One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all. Fluids

The thickness that goes both ways

Heat a liquid and it thins; heat a gas and it thickens. The two are not a strong effect and a weak one but opposite signs, differing by a factor of thirty in size as well — and the word viscosity names one measurement made on two mechanisms that have almost nothing in common.

 Nine hundred steps and hardly anywhere. On the left, a walk of 900 steps of unit length in uniformly random directions, which is what a photon does inside a star: it goes a mean free path, scatters, and starts again in a direction that has forgotten the last one. After 900 steps it is 7.5 lengths from where it began, against 900 if it had gone straight. On the right, the root-mean-square distance over 240 independent walks against the number of steps, both logarithmic: a straight line of slope 0.4920 against an exact one half. The square root is the whole of the result and it is brutal. Escaping a body of radius R takes not R/λ steps but (R/λ)² of them, so a mean free path a thousand times smaller costs a million times as long. That is the difference between a photon leaving the Sun's core and a neutrino doing it: one takes a hundred thousand years and the other takes two and a third seconds, through the same material, and the only thing that differs is λ. What the picture cannot show is the sense in which the escaping energy is not the photon that started: it is absorbed and re-emitted countless times, at falling temperature, so what leaves is a gamma ray's worth of energy arriving as a great many visible photons. Astrophysics

The light that takes a hundred thousand years to leave

A neutrino made in the Sun's core is at the surface in two and a third seconds. A photon made beside it takes something like a hundred thousand years, through the same material, over the same seven hundred thousand kilometres — and the whole of the difference is one length, entering the answer squared.

 The temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative. Thermodynamics

Hotter than any temperature there is

A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

 A wave that dies with nothing to rub against. The electric field of a plasma wave at kλ = 0.5, against time in plasma periods, on a logarithmic scale, obtained by integrating the collisionless kinetic equation as an initial-value problem. There are no collisions in the equation, no viscosity and no resistance; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field nevertheless falls exponentially, at 0.1534 per plasma period, against the published root of the kinetic dispersion relation at this wavenumber, 0.1534, and Landau's asymptotic formula's 0.1514. Meanwhile the free energy of the perturbation — the weighted norm of the distribution plus the field energy, which the equation conserves exactly — moves by 2.4e-10. So nothing has been dissipated: every joule the field loses is still in the distribution, and the accounting closes to a part in ten thousand million. The energy has gone into the particles' ordered motion, and the information about the wave is wound into structure at finer and finer scales in velocity. Astrophysics

The wave that dies with nothing to rub against

Every damping in this collection so far removes energy from a wave and puts it somewhere warmer. This one removes it and produces no heat at all: there are no collisions in the equation, the entropy is unchanged, the whole thing runs backwards perfectly, and the wave still dies exponentially. What it dies into is structure in velocity too fine for a field to see.

 Two quantities that are never computed and never change. A two-dimensional field integrated for 110 steps using only the two equations that contain a time derivative — Faraday's and Ampère's. The two that do not, Gauss's law for the electric field and the statement that there are no magnetic charges, are never imposed and never checked during the run. Their residuals are plotted: the divergence of B stays below 2.4e-16 of the field's own size and the divergence of E below 2.4e-16, over the whole run, while the field itself moves and changes by a factor of 8.59. That is not a numerical coincidence. Taking the divergence of Faraday's law gives the divergence of a curl, which vanishes identically, so ∂(∇·B)/∂t is zero whatever the fields are doing; the same manoeuvre on Ampère's law gives ∂(∇·D)/∂t = −∇·J. So the two constraints are initial conditions, propagated for ever by the two that are laws of motion, and Maxwell's four equations are two dynamical ones and two statements about how the field was set up. Electromagnetism

The two equations that are not laws of motion

Maxwell's equations are usually presented as four laws of equal standing. Two of them contain no time derivative at all, which means they cannot be evolution equations: they are conditions on the field at one instant. What makes them consistent with the other two is that the other two preserve them exactly — and one of the two preservations holds only because charge is conserved.

 Two experiments, two computations, one coefficient. The Seebeck coefficient of a resonant conductor against where its resonance sits relative to the chemical potential, together with the Peltier coefficient divided by the temperature. The first is obtained by applying a temperature difference and finding the voltage that stops the current; the second by applying a voltage at uniform temperature and taking the ratio of the heat flow to the current. Different driving, different measurement, different integral — and the two curves agree to 3.8e-7 of the sweep's own scale across the whole of where they pass through zero and change sign. That equality is Kelvin's relation Π = ST, guessed in 1854 from an argument its author knew was not sound and proved by Onsager in 1931 from microscopic reversibility. It is not a property of this conductor; it holds for every one. Thermodynamics

The second experiment that cannot disagree

Heat one end of a wire and a voltage appears across it. Pass a current through the same wire at uniform temperature and it carries heat. Those are two different experiments with two different apparatus, and the coefficient in front is the same number in both — not approximately, and not for some materials. The reason is that the equations of motion underneath look the same run backwards.

 Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them. Waves

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

 The year, arriving underground at four different times. Temperature against depth in soil of diffusivity 0.5 mm²/s, at four times of year, measured from the annual mean. The surface swings by ±12 K; the swing underground is smaller and later, and both are governed by one length, δ = √(2D/ω) = 2.24 m. The amplitude falls as e^(−z/δ) — the dashed envelope — and the phase lags by z/δ radians, so the curves lean over as they go down and cross the axis at different depths. At 7.0 m the lag is half a year: the ground there is at its coldest in August and its warmest in February, in antiphase with the sky, with a swing of 0.52 K left. That is a cellar, and it is also why a water main below about a metre and a half does not know that it froze last week. Thermodynamics

The summer that reaches the cellar in December

Drive the diffusion equation at its boundary instead of releasing something into it and the solution is a decaying, lagging oscillation with a single length in it. That length governs both the shrinking and the delay, which is why the depth at which the ground is coldest in August is fixed by the same number as the depth at which the seasons stop being felt at all.

 The two holes a grain can fall through, and their exact sizes. Three equal spheres in contact, and four, drawn with the largest sphere that passes between them. The numbers are geometry and nothing else. Three mutually touching spheres put their centres on an equilateral triangle of side 2R, whose circumradius is 2R/√3, so the gap admits a sphere of radius 0.154701R — about a seventh. Four in a square admit 0.414214R, nearly half. A real packing contains both arrangements and everything between, so a grain smaller than the first threshold gets through everywhere, one larger than the second gets through nowhere, and one in between percolates slowly through the loosest routes. That is the whole size-dependence of segregation by percolation, and it is why the effect is reliable below about a seventh and erratic between a seventh and a half. Fluids

The big one comes to the top

Shake a jar of mixed grains and it sorts itself, which is the opposite of what shaking a mixture of gases does. There is no thermodynamic paradox in it because there is no temperature to speak of — and the mechanism is a piece of geometry with an exact number in it: a sphere fits through the gap between three touching spheres only below a radius ratio of 0.1547.

 Runs that break the second law, and how often. The work done in a process repeated many times, and the same for the process run in reverse with its work reflected, for a free-energy change of 4 kT and a dissipation of 3 kT. The average work exceeds the free-energy change, which is the second law, and individual runs do not have to: the shaded tail is the fraction of runs that do less work than the free energy — trajectories in which the entropy of the universe went down — and it is 11.03% here. The two curves cross exactly at the free-energy change, whatever the dissipation, which is what makes an irreversible measurement able to report an equilibrium quantity. Thermodynamics

The second law, with a probability attached

Entropy increases, on average. For a small system pulled quickly, individual runs go the other way — and how often is not a matter of taste but an exact number, fixed by a relation with no adjustable constant in it and no requirement that anything be near equilibrium.

 Colder the bigger it is. The Hawking temperature against mass, on logarithmic axes, with the microwave background drawn across it. The slope is minus one exactly, so a heavier hole is colder — a negative heat capacity, which is the fact everything else here follows from. The two lines cross at 4.50e+22 kg, about a hundredth of the Moon's mass. Anything heavier than that is colder than the sky it sits in and absorbs more than it emits, so it grows rather than evaporates. A stellar-mass hole is at 6.2e-8 kelvin and will not begin to lose mass until the background has cooled below that, which takes something like 10¹² years. Evaporation is not something happening now to any hole anybody has observed. Astrophysics

The hole that outlives everything and then does not

A black hole radiates at a temperature that rises as it shrinks, so losing energy makes it lose faster. The whole history follows from that one sign: a life proportional to the cube of the mass, nearly nothing happening for almost all of it, and an end that arrives in a second.

 The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off. Mechanics

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

 One temperature, four pawls, and nothing gained. The net rate of a ratchet whose gas and whose pawl are at the same temperature, against the load, for notches 2, 5, 10, 20 times the thermal energy deep. Every curve passes through zero at zero load and is negative everywhere else. The device is not merely unable to lift a weight; under any load at all it turns the wrong way and lets the weight down, converting its potential energy into heat in the gas. Making the notch deeper slows everything down — an exponential in the depth — and does not change the sign anywhere. That is the second law arriving as a mechanism rather than as a prohibition. Nothing was assumed about entropy; the pawl was simply allowed to be as warm as everything else, and its own fluctuations undo exactly the rectification it was there to provide. Any rectifier small enough for thermal noise to matter has this problem, and the rectifier being clever does not help, because the same noise reaches the cleverness. Thermodynamics

The engine a fluctuation cannot run

A ratchet lets a shaft turn one way and not the other. Put a paddle in a gas on the same shaft and molecular collisions appear to become a lifted weight — an engine running on one reservoir. It does not work, and following exactly why turns the second law from a prohibition into a mechanism: the pawl is as warm as the gas, and it lifts whenever it is asked to.

 The quantity that went down, and the one that went up. The books for GW150914: two black holes of 36 and 29 solar masses merging into one of 62, with a final spin of 0.67. 3.0 solar masses left as gravitational waves, so the mass fell by 4.6 per cent. The total horizon area rose, from 107417 to 168330 in units of the Sun's gravitational radius squared — an increase of 57 per cent. The two progenitors are taken as non-spinning, which is the assumption that makes the test hardest to pass: a spinning hole of the same mass has a smaller horizon, so any spin they actually had would only widen the gap. Mass is the quantity that behaves the way energy usually does and it is not the one with a direction. Area is, and it is the reason the area has been read as an entropy ever since. Astrophysics

The area that is not allowed to shrink

Two black holes merge and the result weighs less than the sum, because three solar masses left as gravitational waves. The horizon area went up by more than half. Mass is the quantity that behaves like energy and it is not the one with a direction; area is, and the theorem saying so has been tested against a real merger.

 A hundred thousand taps, and still not finished. The packing fraction of a column of grains against the number of taps it has been given, on a logarithmic horizontal scale, for several tap intensities. The grains start where pouring leaves them, around 0.55, and climb towards something near 0.64. At an intensity of 1.2 the packing reaches 0.6318 after a hundred thousand taps; At an intensity of 2 the packing reaches 0.6327 after a hundred thousand taps; At an intensity of 3 the packing reaches 0.6338 after a hundred thousand taps, which is still 0.0097 short of the asymptote. The shape is what matters. On a logarithmic axis the curve is close to a straight line over four decades, which means the packing improves by about the same amount for each factor of ten in the number of taps — not for each additional thousand. Going from a hundred taps to a thousand buys as much as going from a thousand to ten thousand. An exponential relaxation is over after a few time constants and this is not one. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure. Fluids

The pile that is never finished settling

Tap a jar of grains and it settles. Keep tapping and it goes on settling — logarithmically, so that each factor of ten in the number of taps buys the same small improvement as the last. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure, and the asymptote everyone quotes is a fitted number rather than a measured one.

 Two solutions, and nothing in the equations to choose between them. A spherical pulse leaving a point and a spherical pulse arriving at one, each drawn at three times 0.35, 0.6, 0.85 in units where the speed is one. Both are exact solutions of the same wave equation, which is checked here by differencing the drawn samples twice in space and twice in time and requiring the residual to vanish for each. The one on the left is what is always used; the one on the right is discarded, and the equations do not do the discarding. The incoming pulse grows as it converges for the same reason the outgoing one decays as it spreads — the same energy through a smaller sphere — and it is as consistent with conservation as its mirror image is. Electromagnetism

The solution that is thrown away

Maxwell's equations admit a field that converges on a charge exactly as readily as one that leaves it, and nothing in them prefers either. Retardation is a boundary condition rather than a law. Which boundary condition is right has been argued about for a century, one of the answers makes the arrow of time a property of there being absorbers, and the laboratory version of the question — whether an atom emits at all — has a measured answer that depends on what is listening.

 The barrier a reverse field takes away. The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it. Electromagnetism

Nothing keeps a magnetisation for ever

A magnetised particle sits in a well with a barrier between it and the other direction, and a barrier of finite height is crossed eventually. So remanence has a lifetime, coercivity is a different number depending on how fast it is measured, and a grain below about twenty nanometres of iron forgets within a second at room temperature.

 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. Astrophysics

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.

 A wall pulled away fast leaves the energy behind. The energy of a ball bouncing in a box whose wall is moved, against the length of the box, for 3 wall speeds — each a fraction of the ball's own starting speed — with the adiabatic prediction drawn dashed. Every collision is solved for exactly rather than stepped, so nothing here assumes the wall is slow. Moved slowly, the wall takes energy from the ball at the adiabatic rate, and the energy falls as the inverse square of the length. Moved as fast as the ball is moving, it takes almost nothing: the ball cannot catch a wall retreating faster than it travels, so the collisions stop and the energy stops falling. That is the difference between a gas pushing a piston and a gas expanding into a vacuum, and it is drawn here for one particle. At the end of the range the slowest wall leaves the energy at 0.058 of its starting value and the fastest at 1.000, against an adiabatic 0.065. Mechanics

The wall that moves while the ball is in flight

Energy is conserved because the rules do not depend on the time. Move the walls of a box and the rules do depend on the time, so what is inside gains or loses without limit — and how much depends entirely on how fast. Moved slowly, a wall takes energy at exactly the rate the adiabatic law says; moved faster than the ball travels, it takes none at all, and the same box is a piston or a vacuum according to a speed.

 What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all. Relativity

The count that no observer can disagree about

A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

 Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity. Relativity

The column that is hotter at the bottom

Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

 The column that overshoots its own height. The height of water in a vertical glass tube, as a fraction of Jurin's equilibrium height 2γ/ρgr, against time for tubes of radius 0.2, 0.4, 1, 2 mm, integrated from the capillary pull, the weight and the Poiseuille drag with the inertia of the rising column. The 0.2 mm tube (Jurin height 74.4 mm) creeps up without overshooting; the 0.4 mm tube (Jurin height 37.2 mm) creeps up without overshooting; the 1 mm tube (Jurin height 14.9 mm) overshoots to 1.33 of it; the 2 mm tube (Jurin height 7.4 mm) overshoots to 1.46 of it. The linearised motion about the equilibrium is underdamped when the radius exceeds (32η²γ/ρ³g²)^(1/5) = 0.48 mm for water: wider tubes ring, narrower ones do not. Fluids

The pore that lifts highest fills slowest

Jurin's law says how high water climbs in a tube and nothing about how long it takes. The rise has three laws in succession — the column's inertia, then its viscous drag, then its weight — and the same narrowness that lifts water highest throttles it hardest, so for any deadline there is a best pore. A wide enough tube does something Jurin's law cannot: it overshoots and rings. And whatever the route, exactly half the energy the surface releases is lost.

 The price of precision never falls below two. For a clock ticking forwards and backwards with a free-energy drop a per step, the product of its squared relative uncertainty and the entropy it has produced, in units of k: exactly a·coth(a/2), whatever the time elapsed. It approaches the bound of 2 only as the driving vanishes and grows as a once the driving is strong; the three clocks compared elsewhere sit at 2.04 (0.5 kT), 2.63 (2.0 kT), 8.01 (8.0 kT). No steady-state process of any design lies below the line at 2 — the thermodynamic uncertainty relation. Thermodynamics

The precision a clock pays for in heat

Anything that counts — a molecular motor stepping along a filament, a chemical oscillator keeping time in a bacterium, a clock ticking on a wall — has to be driven, because a count that is not driven runs backwards as often as forwards. The heat it gives off buys it regularity, and there is an exact floor under the price: the squared relative uncertainty of the count, times the entropy produced, is never less than two in units of Boltzmann's constant.

 Shake more gently and the grains choose a side. The steady share of the grains in each of two shaken compartments joined by a slot, against B, the parameter that grows as the shaking gets gentler. Below B = 4 the only steady state is an equal share, and it is stable. At B = 4 it becomes unstable and two clustered states appear, continuously, one with each compartment full: the fuller compartment holds 0.855 of the grains at B = 5, 0.929 at 6 and 0.979 at 8. Which compartment fills is decided by the first fluctuation. Nothing in the rules prefers either side, and the equal state that the rules would suggest is the one the grains avoid. Fluids

The sand that chooses a side

Divide a box of grains into two compartments with a slot in the wall between them and shake it from below. Shaken hard, the grains share themselves equally, as molecules of a gas would. Shaken a little more gently, they gather in one compartment and leave the other almost empty, and the crowded side is cold and the empty side hot. Nothing sorted them. Every collision between grains loses energy, and a compartment that happens to hold a few more grains loses more of it and stops throwing grains over the wall.

 The heat an eccentricity buys. The power tides dissipate inside Io against the eccentricity of its orbit, from (21/2)(k₂/Q) G M² R⁵ n e²/a⁶ with k₂/Q = 0.015: it grows as the square of the eccentricity. At Io's forced eccentricity, 0.0041, it is 9.3 × 10¹³ W, against about 1.0 × 10¹⁴ W measured from Io's infrared glow — the same to within the uncertainty of both — and against 4.7 × 10¹³ W for the whole of the Earth's internal heat, from a body forty-three times Io's volume. Halve the eccentricity and the heating falls fourfold; take it away and Io would freeze. Astrophysics

The moon kept molten by its neighbours

Io is the most volcanic body known, with hundreds of active vents and lava hotter than any erupting on the Earth, on a moon smaller than the Earth's own and far too small to have kept any heat from its formation. The heat is made continuously, by tides: Jupiter stretches Io, and because Io's orbit is slightly eccentric the stretch changes every day and a half, flexing the rock and warming it. Tides should have rounded the orbit off in a hundred thousand years and switched the heating off. They cannot, because two other moons keep tugging the orbit back into its oval, and the heat Io radiates is paid for by Jupiter's spin.

 Heat that arrives everywhere at once, and heat that waits. The temperature spread from a point of heat released at time zero, in one dimension, at times 0.5, 2, 8 relaxation times, from the diffusion equation (dashed) and from Cattaneo's telegraph equation (solid) with the same diffusivity, distances in units of the signal speed times the relaxation time. At 0.5, the diffusion equation has put 62 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 78 per cent is still in the two fronts; at 2, the diffusion equation has put 32 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 37 per cent is still in the two fronts; at 8, the diffusion equation has put 5 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 1.8 per cent is still in the two fronts. The fronts, drawn as the vertical marks, carry heat outward at the signal speed and decay; behind them the telegraph solution fills in and, after many relaxation times, becomes the diffusion equation's Gaussian. Relativity

The heat that arrives before it could

Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.

 The same paste stopping or running, according to the push. A thixotropic paste, sheared hard and then held under a constant stress: its viscosity against time, on a logarithmic axis, for stresses of 1.5, 1.8, 2.0, 2.4 Pa. Its structure builds at rest and is broken by flow, and the model's critical stress is 1.89 Pa. Below it the paste's viscosity climbs without limit: it slows, its structure builds faster than the slowing flow can break it, and it creeps to a stop, its viscosity growing in the end as the 1.5 power of time. Above it the viscosity falls: it speeds up, breaks its structure faster, and runs at a steady rate. After a thousand seconds the viscosities stand at 514.1, 238.3, 2.0, 1.5 Pa s: stresses a few per cent apart have gone opposite ways — a bifurcation, not a smooth change — and the paste hesitates longest when the stress is closest to the critical value. Fluids

The paste that forgets it was stirred

A jar of clay slurry left on a shelf sets like a jelly; shaken, it pours. That is thixotropy, and the tempting description — its viscosity falls when it is stirred — misses what makes it strange. A thixotropic paste has no viscosity at a given rate of stirring until it has been told how long it has been stirring, and under a steady push it cannot settle on a slow flow at all. Push slightly less than a critical stress and it creeps to a halt; push slightly more and it runs. Between the two there is nothing, and a paste made to flow slowly splits into a layer that flows and a layer that does not.

 A chain in which every domino is half as large again. The first 6 dominoes of a chain in which each is 1.5 times the size of the one before, drawn to scale, each standing 0.5 of its own height from the next, the first tipping onto the second. The sixth is 7.6 times the first's height and 3325 times its barrier. Each domino falls onto one it could not have been pushed over by hand at the same effort, and the chain passes on not the push but the permission to release energy each domino already held. Mechanics

The push that pays only for the hill

A domino can knock over one half as large again as itself, and that one a still larger one, so that thirteen steps take a fall that began with a fingernail's flick to a slab as tall as a person. Nothing is being amplified by a machine. Each domino holds energy it was given when it was stood on its end, far more than it takes to tip it, and the push that starts it pays only for the small hill in front of that store. A chain of dominoes passes along not energy but permission — which is how a spark lights a fire and a nerve carries a signal the length of an arm without fading.

 The race to equilibrium that the hotter start wins. How far each system is from equilibrium with the bath — the relative entropy of its distribution from the bath's, in nats, on a logarithmic axis — against time since it was placed in the bath at 0.30, also logarithmic, for a start at 0.50 (warm, dashed) and one at 0.901 (hot, solid). The hot system starts further away: 1.243 against 0.254. Both close in quickly as their left wells equilibrate internally. Then the warm system must wait for probability to leak between the wells, and stalls, while the hot one, whose wells already had the right shares, carries on falling. The hot curve overtakes the warm one at a time of about 6.9, and from then on the hotter start is always nearer equilibrium. Thermodynamics

The hotter start that cools first

Put two identical systems, one warm and one hot, into the same cold bath, and common sense says the warm one reaches the bath's temperature first, since the hot one has further to go. The claim that hot water sometimes freezes before warm water — the Mpemba effect — has been argued over for sixty years and does not survive careful measurement. But a cleaner version of it is real and has been observed: in systems whose relaxation has a slow route and a fast one, a hotter start can miss the slow route entirely and overtake a cooler start, exponentially. Whether a system cools quickly depends not on how far it is from equilibrium but on how much of the slowest way back it needs.

 The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper. Waves

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

 The temperature through growing lake ice. Temperature against depth below the ice surface in a lake under air 10 K below freezing, after 1, 4 and 16 days, from Stefan's law with the ice's top held at the air's temperature. The ice is 11.1, 22.3, 44.6 cm thick; within it the temperature rises in a straight line from −10 °C at the top to 0 °C at the bottom, and below it the water is at 0 °C throughout. Ice forms only at the bottom, where water meets ice at the freezing point, and the latent heat it releases there has to be carried up through all the ice above by conduction. The straight lines get shallower as the ice thickens: the same temperature difference drives heat across a longer path, so less heat escapes and the ice grows more slowly. Sixteen times as long gives four times the thickness. Thermodynamics

The ice that grows more slowly the thicker it gets

A lake under air ten degrees below freezing grows eleven centimetres of ice on the first day and only another eighteen in the following six. The ice forms at its underside, and every gram of it releases latent heat that must be carried up through the ice already there before more can form: the ice is its own insulation. The thickness therefore grows as the square root of time, and the rule, Stefan's, depends on temperature and time only through their product — the freezing degree-days that lake-ice forecasters count. A few centimetres of snow on top slows it more than any amount of cold speeds it up.

 A rod in a shear flow turns in fits and starts. The angle between a rigid rod's axis and the flow direction, unwrapped, against the strain of a steady shear flow (shear rate times time), for spheroids of aspect ratio 1, 5, 20 lying in the plane of the shear, each starting along the flow. A sphere (r = 1) turns steadily at half the shear rate. A rod spends most of its time nearly aligned with the flow, turning slowly, and then flips through half a turn quickly; at r = 5 it is within 10° of the flow direction 46 per cent of the time, at r = 20, 82 per cent. Each flip is half of Jeffery's period, π(r + 1/r) units of strain. Fluids

The rod that tumbles for ever in a stream

Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.

 How much of supercooled water freezes in an instant. The fraction of a body of supercooled water that freezes at once when it is nucleated, against how far below 0 °C it had been cooled: the ice that forms releases latent heat until the whole of it is back at 0 °C, and freezing stops there until heat is taken away. Solid: with water's measured heat capacity, which rises steeply below about −20 °C; dashed: with the heat capacity it has at 0 °C. At −15 °C, 19.1 per cent freezes at once; at −20 °C, 25.7 per cent; at −38 °C, the coldest liquid water reaches before it freezes by itself, 51 per cent. The rest stays liquid at 0 °C. Water would have to be cooled about 80 degrees for all of it to freeze at once, and no liquid water survives that long. Thermodynamics

The supercooled water that only partly freezes

Water can be cooled well below 0 °C and stay liquid, and when a crystal finally starts, the whole bottle seems to turn to ice at once. It does not. Freezing releases heat, and with nowhere for that heat to go in the instant of freezing, it warms everything back to 0 °C — at which point freezing must stop. Water cooled to −15 °C freezes about a fifth at once; to −38 °C, the coldest liquid water gets, about half. The rest has to wait for the surroundings to take the heat away, and a hand warmer is the same arithmetic run above body temperature.

 A tilt that reaches zero at a definite time. The tilt of the disk against time from 10°, when the losses are set to end the motion after 100 s, for two ways of losing energy: viscous air squeezed out of the thin wedge under the rim (red), which takes energy faster the smaller the tilt, so that α³ falls in a straight line; and a steady rolling resistance at the contact (blue), which takes energy in proportion to the contact rate, so that α to the power 3/2 does. Both fall slowly at first and steeply at the end, and both reach zero tilt at a definite time rather than approaching it for ever: at 99 s, one second before the end, the tilt is 2.15° under the air model and 0.46° under rolling resistance. The disk's energy, (3/2)Mgaα, runs out in a finite time, and with it the tilt. Mechanics

The disk that spins faster as it stops

A coin spun on a table settles with a whirr that rises in pitch and then stops dead. Everything about it is losing energy, yet the rattle gets faster to the very end. The point where the rim touches the table races round faster and faster as the coin tips flatter, while the coin's face, and the head on it, turns more and more slowly. Two plausible mechanisms of loss both predict that the tilt reaches zero, and the rattle infinity, at a definite instant — a singularity a coin on a kitchen table runs into every time.

 A ring pushed and pulled, and pushed a little more. Over two cycles of the alternating supply, for a ring whose reactance equals its resistance (ωL/R = 1): the flux through the ring (grey), which also sets the radial field that pushes the ring along the core; the voltage induced round the ring (dashed); the current, which lags that voltage by 45° because of the ring's own inductance (blue); and the force along the axis, current times radial field (red). The force pushes the ring away from the coil for part of each half-cycle and pulls it back for the rest; with the lag, the pushes outweigh the pulls, and the average (dotted) is 0.250 of the peak flux–current product. With no lag the pushes and pulls would cancel exactly. The ring jumps because its current is late. Electromagnetism

The ring that jumps because its current is late

Slip an aluminium ring over the iron core of a coil, switch on the alternating current, and the ring leaps off the top. The usual explanation is Lenz's law: the ring's induced current opposes the change, so the ring is repelled. But an alternating field changes both ways every cycle, and an induced current that simply opposed the change would be pushed and pulled equally and stay put. The ring jumps only because its current runs late, held back by its own inductance — which is why a cooled ring jumps far higher and a ring with a saw cut through it does not move.

 The travelling twin's surfaces of now. Surfaces of equal radar time for a twin who leaves at 0.6c and turns round after 4 years of their own time, every half-year of the twin's time from 0.5 to 7.5, drawn in the home frame; each surface is every event the twin would time, by echo, at that moment. On the outbound leg, far from the turnaround's influence, the surfaces are the tilted lines of the outbound rest frame; on the way back, those of the return frame. In between they bend, with a corner where they cross the light rays from the turnaround, and they sweep up the home worldline smoothly: the home clock read at radar times 3.5, 4 and 4.5 is 4.00, 5.00 and 6.00. No two surfaces cross, so no event is given two times, and none is skipped. Relativity

The now a travelling twin can measure

The standard account of the twin paradox has the traveller's sense of “now on Earth” lurch forward by years at the moment of turning round. That lurch belongs to a recipe — borrow the simultaneity of whatever inertial frame the traveller happens to be in — and the recipe is not something the traveller can measure. Timing the home clock by echo, as a radar does, gives a now that the traveller can measure, that never jumps and never crosses itself, and that during the turnaround coincides, surprisingly, with the stay-at-home's own.

 A ring of balls that forgets, and then remembers. Kac's ring: 400 balls on 400 sites round a circle, all white at the start, with 52 of the edges between sites marked; each tick every ball moves one site and changes colour if it crosses a marked edge. The solid line is the excess of white over black, as a fraction of all the balls, against ticks; the dashed line is Boltzmann's prediction (1 − 2μ)^t with μ = 0.1, the fraction of edges marked. For the first few dozen ticks the ring follows Boltzmann's decay exactly as an irreversible process would: the colours mix and the excess falls to zero, then wanders by a few hundredths. At tick 400 every ball has gone once round, through all 52 markers — an even number of colour changes — and the ring is all white again. A rule with no randomness in it, run forward, looked irreversible for 400 ticks and then undid itself. Thermodynamics

The ring that forgets and then remembers

Put a few hundred balls on a ring, all white, and mark a handful of the gaps between them. Each tick, every ball moves one place round and changes colour if it crosses a mark. Nothing is random and everything can be run backwards. Yet the colours mix, half and half, exactly as Boltzmann's equation for a gas says they should, and the entropy rises to its maximum and stays there — until, after one full turn, every ball is white again. Mark Kac built the model in 1956 to show where the arrow of time in Boltzmann's equation comes from, and what it quietly assumes.

 A signal that dies, and comes back when the spins are turned over. The size of the signal from an ensemble of proton spins whose precession rates are spread by 20 Hz (root mean square) about the mean, with a true relaxation time T₂ of 80 ms, against time after they are tipped. The free signal falls to 1/e in about 10 ms and to 0.03 by 20 ms: not because any spin has relaxed but because they have fanned out. A π pulse at 30 ms (marked) turns every spin over, and at 60 ms the signal returns to 0.472 — exactly e^(−2τ/T₂), the dashed envelope, since every static difference in rate has cancelled. Without the pulse the signal at 60 ms is below one part in a million. Quantum

The echo that gives back everything but the wandering

Tip a sample of protons in a slightly uneven magnetic field and their signal dies within milliseconds. None of them has relaxed: they have fanned out, each precessing at the rate its own patch of field sets. Turn every spin over halfway through and the fan closes again, and the signal comes back as an echo, smaller only by the decay that was real. What the echo cannot return is a phase gained while a spin moved, and that one loss, which grows as the cube of the waiting time, is how a scanner finds a stroke within minutes.

 The same damped motion in two phase planes. A cloud of damped oscillators that start on a small circle of positions and velocities, drawn at times 0, 2, 4, 6 (outlines) with the centre's path. Left: position against the physical momentum mẋ. The cloud spirals inward and shrinks: its area falls as e^(−γt), to 0.30 of its start by t = 6, as friction squeezes every motion towards rest. Right: position against the canonical momentum of the Caldirola–Kanai Lagrangian, p = e^(γt)mẋ. The same cloud keeps exactly its starting area while it is stretched out of shape, as Liouville's theorem requires of any Hamiltonian flow. The friction has been hidden in the definition of momentum: the canonical momentum grows exactly as fast as the real one decays. Mechanics

The friction an action can hold only by changing its clock

The principle of least action generates every conservative law of motion from a single integral, and it has no place for friction: there is no ordinary Lagrangian whose equation of motion is a damped oscillator's. Multiply the Lagrangian by an exponential of time and friction appears — but only by redefining momentum so that it grows as fast as the real momentum decays. The phase-space area the principle conserves is then no longer the one that counts motions, and quantised, the damped oscillator becomes more sharply known than the uncertainty principle allows. Friction belongs to a system and the surroundings it loses energy to, and an action for the system alone can only fake it.

 The film that thins too slowly. The thickness of the air film under a silicone-oil drop of radius 0.38 mm pressed onto its pool, against time, on logarithmic axes, by Reynolds's law: dh/dt ∝ −h³, so the film thins quickly at first and then ever more slowly, its thickness falling only as the inverse square root of time once it is thin. The shaded band is one bounce, 4.1 ms, in which the film falls from 10 µm to 1240 nm; the dashed line is the 100 nm at which van der Waals forces pull the surfaces together, reached after 0.65 s. The air does not have to be held up; it has only to be slow to leave, and the cube of the thickness makes it slower and slower the thinner it gets. A vibrating bath that lifts the drop off every few milliseconds refills the film each time. Fluids

The drop that will not merge

A drop of oil let fall onto a pool of the same oil ought to vanish into it at once: merging releases surface energy, and nothing holds the two apart. Instead it bounces. A film of air is trapped between drop and pool, and air, though it offers no strength at all, is slow to leave a gap a few micrometres thick: the film takes the better part of a second to drain to the hundred nanometres at which the surfaces can touch, while a bounce lasts four milliseconds. Shake the pool and the drop can bounce for ever, in rhythms that double as the shaking grows — and, near the pool's own instability, it starts to walk across the surface on the waves of its own making.

 Two metals mixing at unequal speeds. The fraction of the fast-diffusing species A against position across a welded couple, A on the left and B on the right, at three times in the ratio 1 : 4 : 16, when A diffuses 3 times as fast as B, from Darken's equations, in units in which B's diffusivity and the middle time are both one; each profile's marker — an inert wire laid at the original weld — as a dot on it. The profiles spread as the square root of time, and they are lopsided: A pours into B's side faster than B into A's, so the mixed zone extends further into B. The markers do not stay at the weld. They move into A's side — by 0.39, 0.80 and 1.63 units at the three times — because the lattice planes of A's side are being emptied faster than they are filled. That drift, first seen in copper and brass in 1947, is the Kirkendall effect. Thermodynamics

The markers that move because atoms are unequal

In 1947 Ernest Kirkendall and Alice Smigelskas wound fine molybdenum wires around a bar of brass, plated it thickly with copper, and heated it for fifty-six days. The wires, which do not dissolve in either metal, ended closer together: the brass inside them had shrunk. Zinc had diffused out of the brass faster than copper diffused in, and the lattice planes of the brass had been taken away along with the zinc. If atoms in a metal moved by swapping places with each other, the two flows would have had to be equal. They were not, and the only way the books balance is if atoms move by jumping into vacancies — empty lattice sites — that flow the other way. The wires were the first direct evidence that diffusion in metals runs on holes.

 Newcomen's dilemma. Two consequences of how cold a Newcomen engine's injection spray leaves its cylinder, against that temperature: the pressure difference the atmosphere then pushes the piston down with, one atmosphere less the vapour pressure of the water left in the cylinder; and the steam the next stroke wastes reheating the cylinder's iron skin, as a fraction of the most it wastes at 30 °C (8.2 cylinder-fulls). A cylinder cooled to 40 °C gives a push of 0.93 atmospheres and wastes 7.0 cylinder-fulls of steam a stroke; one left at 70 °C wastes half as much but pushes with only 0.69. Every choice was bad, because the one vessel had to be both hot, to receive the steam, and cold, to make the vacuum. Newcomen's engineers ran it in between. Thermodynamics

The cylinder that must not be cooled

The first steam engines that did useful work, Thomas Newcomen's pumps of 1712, made a vacuum by spraying cold water into a cylinder full of steam. It worked, and on every stroke it cooled the cylinder, so that the next charge of steam condensed on the cold iron until the walls were hot again — several cylinders' worth of steam spent heating metal that the next spray would cool. James Watt saw in 1765 that the vessel was being asked to be hot and cold at once, and moved the cold to a separate condenser. The improvement was a factor of three or four in coal. A century later the reason had a name: heat passing from hot to cold across a large difference, doing nothing on the way, is work thrown away.

 A lake a little under half full of gas. Carbon dioxide dissolved in the water of Lake Nyos in Cameroon against depth, in a representative profile of the kind measured in the years after 1986, beside what water at that depth could hold under Henry's law, which rises with the pressure. Everywhere the water holds less than it could: 0.35 mol/L at 200 m, where 0.75 mol/L would be saturated, so the deep water is not on the point of doing anything. It can sit there for decades, fed by a vent in the bed below. But it is only stable where it is. Move a parcel of it up, keeping its gas, and the pressure falls while the gas stays. It meets the saturation line at 88 m, a little over a hundred metres above where it started. Fluids

The lake that holds its gas until it does not

On the night of 21 August 1986, Lake Nyos in Cameroon released about a cubic kilometre of carbon dioxide in a few hours, and some 1,700 people died in their sleep in the valleys below. The lake was not boiling and it was not erupting. Its deep water had been holding the gas safely for decades, in a layering that was stable as long as nothing moved. Lift that water far enough and it cannot stop rising. The same instability, in a pipe, now drains the danger away.

 A cooling planet, read from the top. Temperature against depth in a body cooling by conduction from a uniform start at 3870 °C, with a surface held at nought, at four ages: the exact error-function solution for a half-space. Cooling reaches downward as the square root of time, so after a hundred million years the upper few hundred kilometres have cooled and the interior knows nothing about it. The gradient at the surface is what a mine measures, and it falls as the inverse root of time. A thermal gradient therefore gives an age, and this is what Kelvin saw in 1862. He assumed the Earth was a solid conductor from the start, on a good basis, as everyone believed then, and these curves are the mathematical consequence of that belief. Thermodynamics

The age a cooling Earth gives

In 1862 William Thomson, later Lord Kelvin, took a thermometer reading from the bottom of a mine and a piece of mathematics from Fourier and computed the age of the Earth: about a hundred million years. Geologists, needing far longer, had no physics with which to reply. The arithmetic was right. The usual story blames radioactivity for overturning it, but half the heat is not radioactive, and the error lay in assuming the inside of the planet stays still.

All themes