Illustrated Physics
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Order out of the random — part 2 of 2

Enormous numbers of unpredictable particles producing quantities you can print on a dial.
The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it. Fluids

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

 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.

 A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks. Fluids

The reflection that changes the wavelength

An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.

 A line is what two bodies give. The electron energy spectrum of tritium, against the vertical line a two-body decay would produce. A nucleus emitting one particle has no choice about how to share the energy: momentum conservation fixes it, and every electron comes out at the same energy. The observed spectrum is a continuum running from zero to the full available energy, with a mean at 0.31 of the endpoint. The shape is a count of the ways the energy can be divided between an electron and something else, and the something else was proposed for no other reason than that the shape requires one. Quantum

The energy that did not all arrive

A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.

 Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore. Quantum

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

 Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds. Astrophysics

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

 A hill that is always ahead of the atom. The light-shift potentials of the two ground sublevels of a spin-½ atom in two counter-propagating beams with crossed linear polarisations, over one and a half wavelengths, in units of the well depth. The polarisation of the light turns from σ+ to linear to σ− every quarter wavelength, and the two sublevels see sinusoidal potentials a quarter wavelength out of step. Optical pumping transfers the atom from one sublevel to the other fastest exactly where its own potential is highest — the fastest pumping and the hilltop coincide, which the figure checks — and that is the bottom of the other potential. The heavy line is an atom that starts with 3.3 well depths of kinetic energy: each time it reaches a hilltop it is pumped down by exactly one well depth, climbs the next hill, and is pumped down again, 3 times, until it no longer has the energy to reach a hilltop and is left oscillating in one well. The energy is carried off by the pumping photon, which leaves bluer than the light that drove it by the depth of the well. Astrophysics

The limit that belonged to a simpler atom

The theory of laser cooling predicted a floor, and the first careful measurement came in six times below it. Nothing was wrong with the measurement or the arithmetic. The floor belonged to an atom with one ground state, and real atoms have several — which lets the light build a hill in front of every atom, move it to the bottom before it can roll back, and repeat the trick until the atom is a few microkelvin from rest.

 A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device. Optics

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

 Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it. Fluids

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

 Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start. Mechanics

The calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.

 Light that walks through a cloud. Seven photons entering a slab 8 scattering mean free paths thick from above, straight down, and followed until they leave, with every scattering equally likely to send them in any direction; their paths are projected onto the page. Of 20000 photons followed the same way, 82.5 per cent come back out of the top and 17.5 per cent out of the bottom. Diffusion theory predicts 18.2 per cent through the bottom. Only 10 photons cross without scattering at all, where e⁻⁸ predicts 6.7 — within the counting error of so few: nearly all of what is transmitted has walked, and the direction it arrived from is forgotten on the way. Optics

The cloud light has to walk through

A beam crossing thirty scattering lengths of anything should keep e⁻³⁰ of itself — a ten-millionth of a millionth. A cloud thirty scattering lengths thick lets through nearly a third of the sunlight falling on it. The light has not crossed; it has walked, one scattering at a time, and a walk through a slab obeys a law with the shape of Ohm's rather than of an exponential.

 Every reaction's free energy has its lowest point inside. An ideal reaction A ⇌ B at 298 K. Across: how far it has gone, from pure A towards pure B. Up: the Gibbs energy of the mixture per mole, relative to pure A. Left, for standard reaction Gibbs energies ΔG° of −4, 0, +4 kJ/mol: the dashed straight lines are what the energy would be if A and B did not mix, and the solid curves add the entropy of mixing them. For ΔG° = −4 kJ/mol the lowest point is at 83.4 per cent B; for ΔG° = 0 kJ/mol the lowest point is at 50.0 per cent B; for ΔG° = +4 kJ/mol the lowest point is at 16.6 per cent B. Right, magnified near pure A, a reaction with ΔG° = +10 kJ/mol, whose straight line climbs from the start and which looks as if it should not proceed at all: its curve first falls, to a minimum of −43 J/mol at 1.74 per cent B, because the mixing term falls infinitely steeply away from a pure end. Each minimum was found by search and sits where the ratio of B to A equals exp(−ΔG°/RT). Thermodynamics

The reaction that cannot go all the way

Chemistry speaks of reactions that go to completion and reactions that do not happen, and at equilibrium there are neither. The reason is a logarithm. The free energy of a half-finished reaction contains the entropy of mixing, whose slope is infinite at both pure ends, so every reaction's lowest point lies strictly inside — and the slope of that free energy, the chemical potential, is to particles what temperature is to heat.

 The light of a diode at room temperature is as bright as a surface thousands of kelvin hot. How many photons occupy each mode of the light, on a logarithmic scale, against photon energy. The lowest curve is the thermal glow of a 1.42 eV semiconductor at 300 K with no voltage across it. The solid curve above it is the same device with 1.3 V across it, emitting only above its gap. The dashed curve is a blackbody at 2571 K, the temperature whose light has the same occupation as the diode's at 1.472 eV, just above the gap. They cross there and nowhere else: the diode's occupation falls a factor e every 25.9 meV, as its lattice's temperature requires, and the blackbody's every 222 meV. No single temperature describes the diode's light. At each photon energy it has a brightness temperature, and that temperature is 300 K multiplied by ε/(ε − qV). Thermodynamics

The glow that carries a voltage

Thermal radiation has no chemical potential, because walls make and destroy photons freely. A light-emitting diode is a body that glows at room temperature with a voltage written into its light — Planck's law with the voltage as the photons' chemical potential — which is why its light can be as bright as a surface thousands of kelvin hot, why at low voltage it can put out more light than the power it draws and cool itself doing so, and why a reverse voltage makes a surface look colder than it is.

 With interference kept, transmission falls exponentially; without it, only as one over the thickness. Light through stacks of randomly thick transparent layers, alternating indices 1 and 2.6, with thicknesses scattered by ±50 per cent about a quarter wave, on a logarithmic scale against the number of layers. The falling line is the transmission with the waves' interference kept, computed exactly by multiplying transfer matrices and averaged as the logarithm over 40 random stacks and five wavelengths. It falls in a straight line: the transmission drops by a factor of e every 17 layers, however thick the stack, which is exponential decay — localisation. The upper curve is the same stacks with every surface's reflection and transmission added as intensities, so that no interference survives. It falls only as one over the thickness, checked to grow in exact proportion, which is the diffusion of light through a cloud: at 400 layers it still transmits 1.0 per cent where the coherent stack typically transmits 4.5·10⁻¹¹. Optics

The walk that interference can stop

Light scattered many times walks through a cloud, and a walk always gets through eventually — a slab twice as thick lets through half as much. Keep the waves' interference instead of adding intensities, and in one dimension the same disorder does something a walk cannot: it stops the light exponentially, traps it in modes with nothing special about where they sit, and turns transmission from a number into a spread over powers of ten. Whether the same can happen to light in three dimensions has been claimed, retracted and argued for thirty years.

 The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150. Astrophysics

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

 Mean field always pushes; correlations pull. The pressure between two planes of equal charge with only their own counterions between them, against their separation, in units of the Gouy–Chapman length μ for the separation and 2πℓ_Bσ²kT for the pressure. The upper curve is the Poisson–Boltzmann result, solved from k·tan(kd/2) = 1: it is the density of counterions at the midplane and is positive at every separation, falling from the ideal-gas 2/d at contact to π²/d² far apart — 1.71 at 1μ, 0.290 at 4μ. The lower curve is the strong-coupling limit, 2/d − 1, which the same ions reach when their valence and the surface charge are high. It crosses zero at d = 2μ and is negative beyond, tending to −1: the two like-charged planes attract, and the separation 2μ is where they come to rest. Fluids

The like charges that pull together

Two surfaces carrying the same charge, with nothing between them but the ions that neutralise them, ought to repel, and the standard mean-field theory proves that they always do. With calcium or spermine as the counterions they attract, and come to rest a fraction of a nanometre apart. The mean field misses it because it averages the ions into a smooth cloud, and multivalent ions are too strongly repelled by each other to form one. Each keeps a patch of surface to itself, and the pressure between the plates becomes a single ion's business.

 A gas of grains that lose 51 per cent of their energy per collision, cooling into clumps. 2000 discs in a square box with periodic walls, covering 25 per cent of its area, started with random velocities and left alone. Each collision keeps a fraction e = 0.7 of the relative velocity along the line of centres. The panels are the same gas at three temperatures, where temperature means the mean kinetic energy of a grain: 0.85 T₀, 0.019 T₀, 0.00049 T₀. It starts uniform — the grain counts in a 10 × 10 grid have a variance 0.3 times their mean, more even than scattered points because discs cannot overlap — and ends in dense bands and clumps with empty space between, the variance 17 times the mean. Nothing attracts anything. The clumps form because a dense patch collides more often, cools faster, loses pressure and is squeezed denser by the hotter gas around it. Fluids

The gas that cools itself into clumps

Shake a box of grains hard enough and they fly about like the molecules of a gas. Stop shaking and the gas cools, because every collision destroys a little of the motion — and it does not cool evenly. A denser patch collides more often, cools faster, loses pressure, and is squeezed denser by the hotter gas around it. With nothing attracting anything to anything, the gas gathers itself into clumps and bands, and in the limit three grains can collide infinitely many times in a finite time.

 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.

 The thin parcel that keeps sinking. A parcel pushed one centimetre down a column in which heat stabilises and salt destabilises, with a density ratio of 2, integrated for three hours, for parcels whose pattern of sinking and rising water repeats every 2 m, 20 cm, 6 cm. The 2 m parcel keeps its heat and its salt and swings back and forth with the buoyancy period of 23.6 minutes. The 20 cm parcel crosses its starting level 8 times while its swing dies away, and is left sinking slowly, at the 0.48 e-folds an hour its width allows; the 6 cm parcel never comes back and passes five centimetres below it after 60 minutes. The thinnest loses its heat to its surroundings in 11 minutes and its salt in 18 hours, so it arrives salty and cooled, heavier than the water around it, and runs away at 2.33 e-folds an hour — the rate the linear dispersion relation gives for that width. Fluids

The salt that sinks through a stable sea

Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.

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

The wrong orbit beside a right one

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

 Hydrogen comes apart far below its binding temperature. The fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one. Thermodynamics

The electron that leaves because there is room

It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of.

 A field strings a suspension into chains. 48 identical polarisable spheres, two radii apart at least, scattered at random through a periodic cell twenty radii square — 38 per cent of its area — and the same spheres after the field (vertical) has been on long enough for the viscous liquid to let them move. Each moves in proportion to the dipolar force from all the others and is stopped only by contact. At the start no two are touching; at the end 45 are bonded end to end along the field and 48 of the 48 spheres belong to chains, lying along the field lines and kept apart sideways because side-by-side dipoles repel. Chains that reach from one electrode to the other are what make the liquid resist shear, and they form in the milliseconds it takes a sphere to travel its own diameter. Electromagnetism

The liquid a field turns solid

Stir fine starch or glass beads into silicone oil and the result pours like cream. Put a few kilovolts across it and within milliseconds it holds its shape and will not flow until it is pushed hard. Nothing in it is charged. Each particle becomes a dipole in the field, dipoles pull on each other end to end and push apart side by side, and so the particles string themselves into chains from one electrode to the other — a solid made of forces that exist only while the voltage is on.

 Flow curves that bend back, and one that stops. Shear stress against shear rate, both on logarithmic axes, for volume fractions 0.5, 0.55, 0.565, 0.575, 0.6, from the same model: stress in units of σ, rate in units of σ over the liquid's viscosity. Below φ = 0.569 every curve rises steadily — the suspension thickens but continuously. Above it and below the frictional jamming point 0.58 the curve bends back into an S: over a range of stress, more stress gives less flow, and a suspension driven at a fixed rate must jump between branches. At φ = 0.6 the curve returns to zero rate at 2.47 σ*: pushed that hard, the suspension stops flowing altogether and holds the stress like a solid. Fluids

The suspension that seizes when pushed hard

Cornflour stirred into half its volume of water runs off a spoon like cream, and a fist driven into it bounces off as if it had hit a floor. The particles have not changed and neither has the water. What changes, above a threshold stress, is the kind of contact between neighbouring grains: lubricated, they slide past one another and the suspension can pack to 64 per cent before it locks; pressed into frictional contact, they lock at 58. A suspension sitting between those two numbers is a liquid when handled gently and a solid when handled hard.

 The same rate for every size of molecule. The diffusion-limited rate constant for two equal spherical molecules meeting in water at 25 °C, against their radius from 0.1 to 100 nm: 8kT/3η = 7.43·10⁹ per molar per second, flat. The dashed line is what a larger target would give if its partner kept the mobility of a 0.3 nm molecule — 7.4·10¹¹ at 30 nm — the intuition that a bigger molecule is easier to hit. Stokes and Einstein's law takes it away again: a molecule twice as big is twice as easy to hit and diffuses half as fast, and the two cancel exactly. Every reaction whose partners react at the first touch runs at close to this one number, whether the partners are ions, small organic molecules or proteins. Thermodynamics

The speed limit that does not care how big the molecules are

Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.

 A tiny ripple that grows, peaks, and gives the energy back. The highest point of a wave train's envelope against time, starting from an even train of amplitude 1 with a ripple of 1 per cent at the fastest-growing wavenumber, integrated from the nonlinear Schrödinger equation. For a while nothing happens that the eye could see; the ripple grows exponentially from a per cent. Then, near t = 5.6, the energy of a whole modulation period gathers into one crest 2.41 times the original amplitude — 1 + √2, the height of the Akhmediev breather this wavenumber produces. Then it spreads out again, and by t ≈ 11 the train is almost even once more, its highest point 1.01: the energy returns to where it came from, and the cycle repeats, the crest rising again near t = 17. The wave's total norm is conserved throughout to a part in a hundred million. Waves

The even wave train that cannot stay even

In the mid-1960s a team at the National Physical Laboratory tried to make a long, perfectly regular train of waves in a towing tank, and could not: however carefully they set the paddle going, a few hundred metres down the tank the waves had bunched into groups. Their paddle was not at fault. A uniform train of steep waves on deep water is unstable — small modulations of its height grow, the energy of many waves gathers into a few, and then, astonishingly, gives itself back. Out of that instability comes a wave three times the height of the sea around it, with no history before and none after.

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

The diffusion that slows the longer it is watched

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

 Why the mixture of two equal waters is heavier than either. A temperature–salinity diagram for seawater near the surface, with lines of equal density (curves) 0.3 kg/m³ apart. Two waters lie on the same line: one at 4 °C and 33.93 g/kg, the other at 16 °C and 36.53 g/kg, both 1026.00 kg/m³. Mixing them in any proportion moves along the straight line between them, because temperature and salt add linearly; the lines of equal density curve, because cold water's density responds less to temperature than warm water's. The straight line lies on the dense side of the curve. An even mixture, at 10 °C and 35.23 g/kg, is 0.182 kg/m³ denser than either parent, and sinks between them. Fluids

The mixture heavier than either water

Two bodies of seawater can have exactly the same density — one cold and fresh, one warm and salty — and sit side by side without either wanting to sink. Mix them, and the mixture is denser than both, and sinks. Nothing has been added and nothing removed; the extra weight comes from the fact that water's density is a curved function of temperature while mixing averages temperature in a straight line. The same curvature, at its most extreme, is why a lake overturns twice a year and freezes from the top, and why the open ocean, too salty to have a density maximum above freezing, does neither.

 How much darker a surface looks under a film of water. The diffuse reflectance of a surface covered by a thin film of clear water (solid), against its reflectance dry, with the dry value for comparison (dashed): (1 − Rₑₓₜ) a (1 − Rᵢₙₜ)/(1 − a Rᵢₙₜ), where the film traps and returns light that falls outside its escape cone. A bright surface of reflectance 0.9 falls to 0.77; a mid-grey one of 0.5 to 0.32, 64 per cent of its dry value; a dark one of 0.2 to 0.11, 54 per cent. Dark surfaces darken proportionally most, because each return trip costs them most; and a coloured surface grows more saturated, because the colours it absorbs are absorbed again on every return while the colour it reflects survives. Optics

The sand that darkens when it is wet

Wet sand is darker than dry sand, a wet stone darker than a dry one, a splashed pavement darker in patches, and a wet white shirt turns grey and then see-through. Water is clear and absorbs almost nothing, so it cannot be adding darkness. What it does is change where the light goes. A film of water on a surface traps light that has been scattered back up, returning most of it for another chance to be absorbed. Water in the pores between grains replaces air with something closer to the grains' own refractive index, so each grain turns light aside less, and light wanders deeper before it gets out. Both lengthen the path, and on a longer path more light is absorbed by the material that was there all along.

 A cluster grown by random walkers. 3000 particles, each released far away to wander at random on a square grid until it first touches the cluster and sticks there, starting from one seed at the centre. Colour marks the order of arrival: the first third (violet), the second (blue), the last (red). The cluster is 220 grid steps across, branched and open, and almost every late arrival is at the outer tips: walkers wander into the outer branches long before they can find their way into the gaps between them. The gaps never fill. Thermodynamics

The cluster that grows where the walkers arrive

Let particles wander in one at a time from far away, each sticking where it first touches a growing clump. The rule has nothing in it about shape, and the clump that results is a lacy, branched thing whose mass grows more slowly than its area, so it gets emptier the bigger it gets. The reason is that a random walker reaches the outermost tips long before it can find its way into a gap between branches. The same arithmetic — the probability that a walker arrives is a solution of Laplace's equation — shapes electrodeposits, lightning-like discharges in plastic, fingers of water pushed into oil, and colonies of bacteria short of food.

 A pile of glass plates, added two ways. The transmission of light at normal incidence through a pile of N glass plates (index 1.5) in air, on a logarithmic axis, against N. Dashed: Stokes's 1862 result for light whose reflections add as intensities, T = (1 − R)/(1 + (2N − 1)R) with R = 0.040 per surface, which falls only as 1/N: 0.107 at 100 plates, 0.029 at 400. Solid: the same pile with the reflections added as waves and every plate and gap of random thickness, as ⟨ln T⟩ over 200 random piles (red) and for one pile (grey). It falls in a straight line, by 0.0843 per plate, an exponential decay with a localisation length of 24 plates: 10⁻³ at 100 plates and 10⁻¹⁴ at 400. Interference with random phases does not average away; it traps. Waves

The pile of glass that stops every colour

A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

 A bounce that turns into a swing and back. The share of a swinging spring's oscillation energy that is in its sideways swing rather than its up-and-down bounce, against time in units of the swing's period over 2π, for a spring started bouncing with a stretch of 0.08 of its length and almost no tilt (0.02 rad). When the bounce frequency is exactly twice the swing frequency (red), the energy drains out of the bounce into the swing until 100 per cent of it is swinging, then drains back, over and over. At a ratio of 2.4 (blue) the swing never takes more than 1.5 per cent. The bounce pumps the swing parametrically, twice per swing, which works only when the two keep step. Mechanics

The spring that swings when it was only bounced

Hang a weight on a spring and pull it straight down: it bounces straight up and down. Unless its bounce happens to be twice as fast as its swing would be as a pendulum. Then the bounce dies away and the weight starts to swing, wider and wider, until the bounce has gone completely — and then the swing dies and the bounce comes back. Two motions that look independent are trading their energy through a resonance, regularly and for ever at small amplitude. Give the spring more energy and the trading becomes chaotic: the same simple rule, with nothing random in it, produces motion that never repeats and that a hundred-millionth of a radian can redirect.

 Too little noise, the right amount, too much. The particle's position over 8 cycles of the push (thin grey line, scaled to the wells), with noise of intensity 0.03, 0.085, 0.3 (top to bottom), the barrier being 0.25. With little noise it stays almost always in its starting well and only trembles: 0 hops in eight cycles. With noise 0.085 it hops 30 times, almost always once per half-cycle and almost always while the push is helping — a clean square wave following a push too weak to cause a single hop. With noise 0.3 it hops 285 times, at random, and the push is lost again. There is a best amount of noise, and it is not zero. Waves

The noise that helps a weak signal through

A system that can sit in either of two states, nudged back and forth by a periodic push too weak to switch it, does nothing at all — until noise is added. Then it begins to switch, and with the right amount of noise it switches in step with the push, once each half-cycle, as though the push had been strong enough all along. Too little noise and it never switches; too much and it switches at random. The response peaks at a nonzero noise, which is why the effect is called a resonance, and why crayfish, paddlefish and human fingertips appear to use it.

 Water drops that charge two cans to a spark. The voltage between the two cans of a Kelvin water dropper against time, on a logarithmic axis, starting from a stray difference of 1 V, for 10, 20, 40 drops a second in each stream; drops 3 mm across, each with a capacitance of 0.17 pF to its ring, cans of 20 pF. Each drop carries a charge induced by the opposite can's potential into its own can, which raises the opposite can's induced charge on the next drop: the difference grows exponentially, multiplying by e every 6.0 s at 20 drops a second. It passes the 10 kV that will jump a few millimetres of air after 55 s — from one volt to a spark, with nothing but falling water. Electromagnetism

The water drops that charge themselves to a spark

Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.

 The peak that rises out of a cloud. A slice through the shadow a released cloud of a million atoms casts after it has expanded, the measurement by which condensates are seen, at 1.1, 0.8 and 0.5 times the transition temperature: the thermal atoms, whose spread of speeds sets the cloud's width, give a broad rounded profile; the condensed atoms, all in one state with only the small spread of momenta their trap allowed, give a narrow peak at the centre, drawn here with the shape repulsion between the atoms gives it. Above Tc there is only the broad cloud. At 0.8 Tc a sharp peak stands up from its middle, holding 47 per cent of the atoms; at 0.5 Tc the peak holds 87 per cent and the thermal wings have shrunk. The two-component shape — a narrow peak on a broad base — was the signature that announced the first gaseous condensates in 1995. Quantum

The condensate a trap makes

A gas of atoms held in a magnetic trap and cooled to a few hundred billionths of a degree does something no gas in a box does at any temperature it can reach: a large share of its atoms falls into the trap's single lowest state, and a sharp peak rises out of the cloud. The reason is counting. The trap's excited states can hold only so many atoms at a given temperature, and the number grows as the cube of the temperature — so below one temperature the atoms run out of room, and every atom that does not fit has only one place to go.

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

The atoms the light stops touching

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

 Two beams that each stay straight, and wind each other into whirlpools. Phase space of a one-dimensional particle simulation: every dot is an electron, plotted at its position along a periodic box one fastest-growing wavelength long (10.26 v₀/ωₚ) and its velocity, in units of the beam speed. One beam starts at +v₀, the other at −v₀, with a 0.0001 ripple in position. At t = 18/ωₚthe beams are visibly rippled; by 26 the ripple has grown into overturning, and by 40 the two beams have wound round each other into a single vortex, filamented into strands. When growth first stops, at ωₚt = 25.2, the wave's field is 0.56 m v₀ωₚ/e; electrons trapped in it bounce at 0.59 ωₚ, faster than the 0.354 ωₚ at which it was growing, and it traps any electron within ±1.92 v₀ of the wave's speed — zero, by symmetry — so both beams are inside its separatrix; total energy was conserved to 0.31%. Astrophysics

The two beams that unravel each other

A cold beam of electrons, alone, is perfectly stable: disturb it and it rings at its plasma frequency and carries the ringing downstream. Send a second beam through it the other way and the pair is not stable at all. A ripple of the right length grows by a factor of e in less than half a plasma period, takes a third of the beams' drift energy into an electric field, and then winds both beams into a single whirlpool in which their drift is gone. Neither beam can do it. The growth exists only in the pair, and a beam a millionth as dense as the plasma it crosses is enough to start it.

 A disc that makes its own field once it turns fast enough. A self-exciting disc dynamo started from rest, its motor's torque set so that, with no current, the disc would run at 0.8, 1.5, 2.5 times the critical speed R/M. Dashed: the disc's speed, in units of the critical speed. Solid: the current in its coil, in units of √(G/M), starting from a stray seed of a millionth, of either sign. Time is in units of the circuit's own L/R. At 0.8 the disc settles at its free-running speed, below critical, and the seed dies. Above it the disc spins up past the critical speed and nothing happens for a while — the current is growing exponentially from a millionth — until it reaches a tenth, at t ≈ 29 for 1.5 and 11 for 2.5 and becomes large enough to brake the disc. Speed and current then ring and settle at exactly the critical speed and a current of ±√(g − 1): 0.71 and 1.22, of the sign the seed happened to have. Astrophysics

The dynamo that reverses with nothing pushing it

A copper disc spinning in a magnetic field drives a current from its axle to its rim. Wire that current through a coil wound round the disc and the coil supplies the field the disc needs — and above one definite speed, a resistance divided by an inductance, the arrangement makes its own field out of any stray one, with nothing else magnetic anywhere. It cannot choose which way the field points; the stray field chooses. Couple two such discs, each feeding the other's coil, and with the motors' torque held perfectly steady the field flips direction again and again at intervals that follow no pattern. The simplest dynamo that works is also the simplest one that reverses.

 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.

 Bands of precipitate laid down in a tube of gel. A tube of gel holding one salt, B, with a second salt, A, diffusing in from its left end, computed at the end of the run: the shading shows where solid precipitate has formed, darker for more. The product of their reaction precipitates only where its concentration first passes a nucleation threshold, 0.5 in the units used, and then wherever solid is already present. The solid is not spread smoothly. It forms in 23 sharp bands, close together near the entrance and further and further apart down the tube — the last two are 26.0 units apart, against 0.8 between the first two. Each band holds more precipitate than the last, in proportion to its distance. Nothing in the gel was patterned to begin with. Thermodynamics

The rings a reaction lays down in a gel

Let one salt soak into a tube of jelly that holds another, and where they meet a solid should form, spreading smoothly behind the advancing front. It does not. It forms in sharp, separate bands, close together near the entrance and further and further apart, each one a fixed multiple of the distance of the last. Raphael Liesegang saw the rings in 1896 and they have been found since in rocks, in agate, in gallstones and in microchips. Nothing in the jelly is patterned to begin with. The pattern comes from a solid that will not start until the solution is overloaded, and that then empties the solution around it.

 A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits. Fluids

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.

 The swing's plane, turning a step at a time. The direction in which a swinging spring swings, seen from above, against time, for a spring whose bounce is twice as fast as its swing, started bouncing almost vertically with a trace of sideways motion. Points are coloured by how much of the energy is in the swing (darker: more). While the swing holds the energy the direction is fixed; while the bounce holds it, the direction changes, and the swing returns in a new direction each time — 9 steps in 1500 units of time, each 43.6° on average (43.6, 43.6, 43.6, 43.6…), always in the same sense. In a plane the swing could only come back along its old line. In space it comes back turned, and the turn is the same every time. Mechanics

The swing that turns a step at a time

A weight on a spring whose bounce is twice as fast as its swing trades its energy back and forth: the bounce becomes a swing, and the swing a bounce again. Let it move in space rather than a plane, with the slightest sideways twist, and each swing comes back in a new direction, turned by the same angle every time. The turn happens all at once, while the weight is only bouncing and the swing is too small to see, and its size is set by a single ratio, of the twist to the energy carrying it. The steps are a geometric phase made mechanical, and they blur into chaos when the spring is bounced too hard.

 Stopped for half a second, a raw egg starts again by itself. Two eggs spinning at 3 revolutions a second, each touched with a finger at t = 4 s so that its shell is held still for 0.5 s and then released. The cooked egg (solid) stops and stays stopped. The raw egg's shell (dashed) stops too, but its contents (dotted) are still turning — slowed to 2.15 revolutions a second during the hold, by viscosity working against the held shell — and once the finger lifts they drag the shell back into rotation, reaching 1.81 revolutions a second, 60 per cent of its speed before the touch. The table's friction is left out here, so that nothing else slows either egg. It is the oldest kitchen test of whether an egg is cooked, and the whole of the physics is a liquid's reluctance to stop turning. Mechanics

The egg that keeps turning after it is stopped

Spin an egg on a table, touch it for a moment to stop it, and let go. A cooked egg stays still. A raw one starts turning again by itself, as if something inside had kept going — which it has. A raw egg is two bodies, a thin shell and a liquid that the shell can turn only through the slow grip of viscosity, and the kitchen test that tells them apart is a measurement of how long that grip takes. The same two-body arithmetic says that a raw egg wastes most of the energy of a flick, starts out spinning six times faster than a cooked one given the same twist, and keeps a record of its past rotation in its interior for a second or two after its shell has forgotten.

 A surface whose thickness depends on how much of it is looked at. The rms roughness of a water surface at 20 °C, in nanometres, against the size of the patch over which it is measured, from the thermal ripples that fit inside the patch — with gravity (solid) and without it (dashed) — for ripples down to a molecular wavelength of 0.6 nm. Over 10 nm the surface is 0.16 nm rough; over a micrometre, 0.26; over a millimetre, 0.36; and beyond a few millimetres gravity flattens the longest ripples and the roughness stops at 0.390 nm. In free fall it would keep growing as the square root of the logarithm: 0.43 nm over a metre. An X-ray reflectivity measurement sees the ripples smaller than its own coherence length, typically some micrometres, and reports a thickness of about a third of a nanometre — a number that belongs to the instrument as much as to the water. Fluids

The surface whose thickness depends on the view

The surface of a glass of water looks like a perfect plane, and over most distances it nearly is. But it is at room temperature, and every ripple it could carry — from a few molecules long to a few millimetres — is shaken by heat with its own share of thermal energy. The ripples are tiny, a fraction of a nanometre high, and they add up in an odd way: every factor of ten in wavelength contributes the same amount of roughness. So the thickness of a water surface has no single value. It depends logarithmically on how large a patch is looked at, until gravity stops it at a few millimetres, and the X-ray measurements that report a third of a nanometre are reporting a property of their own resolution as much as of the water.

 The wait after an eruption, against the eruption. For 400 simulated eruptions of the model geyser: the interval to the next eruption against the duration of the eruption just finished (dots), and the straight line through them, interval = 33.8 min + 10.5 × duration, with a correlation of 0.94. The eruptions fall in two clusters, short ones of about 2.2 minutes followed by waits near an hour and long ones of about 5.0 minutes followed by waits near an hour and a half, and the line runs through both. Its slope is the ratio of the discharge rate to the refill rate, 10.7: each minute of eruption throws out water that takes 10.7 minutes to replace. Plotted against the eruption that follows instead, the same intervals show a correlation of 0.05 — nothing. The interval is set by the past and tells nothing about the future. Thermodynamics

The geyser that remembers one eruption

Old Faithful is famous for being predictable, and its predictability has an odd shape. Its eruptions come in two kinds, short and long, and the wait after each is short or long to match — about an hour after a short eruption, about an hour and a half after a long one. So the next eruption can be forecast from the last, to within about ten minutes, and from nothing else. The reason is a bookkeeping of water. An eruption throws out a volume of water that the geyser must take back from below before it can erupt again, and a long eruption throws out more. The geyser carries a memory of exactly one eruption: how much it has to refill.

 The same grains as a crystal and as a liquid. 160 grains repelling one another by a screened Coulomb force, held together by a bowl-shaped confinement as a levitated layer is, and jostled by gas collisions at two temperatures; each picture is the end of a simulation of their motion. Grains with exactly six close neighbours are coloured. Left: strongly coupled — thermal energy about a thousandth of the interaction at the spacing — the grains settle into a hexagonal lattice, 79 per cent of those away from the edge with six neighbours. Right: the same grains with their thermal energy raised seventy-fivefold, the lattice melted, 9 per cent. In the laboratory the grains are a hundred micrometres apart and big enough to film with an ordinary camera, so the melting of a crystal can be watched particle by particle. Astrophysics

The dust a plasma freezes into a crystal

Sprinkle microscopic plastic spheres into a glowing discharge and each one charges itself to tens of thousands of electrons, because the plasma's light, fast electrons reach it far more often than its heavy ions. The charge holds the grains up against gravity in the sheath above the electrode, and it makes their mutual repulsion so strong compared with their jiggling that, at room temperature, they freeze into a hexagonal crystal with a lattice spacing a camera can resolve. A plasma, the hottest and most disordered state of matter, is where the most strongly ordered one was found to form most easily.

 A chaotic rotor heats without limit, and its quantum version stops. The mean kinetic energy of a rotor kicked once a period by a potential of strength K = 5, against the number of kicks, starting at rest. Classically (an ensemble of 4,000 rotors with random angles) the motion is chaotic and the momentum random-walks, so the energy grows in a straight line, 6.3 per kick, near the K²/4 = 6.25 a random walk of kicks K sin θ gives. Quantum mechanically, with an effective Planck constant ƛ = 1 and 0.5, the energy follows the classical line for a while and then stops, fluctuating about 58 and 123: the smaller ƛ lets it follow the classical growth for longer and climb further before it stops. Nothing in the quantum rotor's equations damps it; the growth stops by interference. Quantum

The heating a quantum rotor refuses

Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.

 How narrow a laser's line can be, against its power. The Schawlow–Townes limit on linewidth, πhνΓ²n/P with Γ the cavity linewidth and n the inversion factor, against output power on logarithmic axes, for a helium–neon laser with a 30 cm cavity of finesse about 300 (Γ = 1.6 MHz) and for a semiconductor laser a third of a millimetre long (Γ about 30 GHz, n = 2), the latter also with Henry's linewidth enhancement 1 + α² for α = 4 (dashed). At a milliwatt the helium–neon laser's limit is 0.0025 Hz — millihertz, a coherence length of tens of millions of kilometres — and the semiconductor laser's 7.2·10⁵ Hz, 1.2·10⁷ Hz with the enhancement. Both fall in proportion to one over the power: more photons in the mode make each spontaneous kick a smaller fraction of the field. The square of the cavity linewidth is why a short, lossy diode cavity has a line ten billion times broader. Optics

The width a perfect laser still has

A laser's light is the nearest thing to a single pure frequency that anything makes, and it is still not one. Every photon spontaneously emitted into the lasing mode adds a kick of random phase, and because nothing in a laser cares what its phase is, nothing pulls the phase back: it wanders, and a wandering phase is a line of finite width. Schawlow and Townes worked out how wide in 1958, before any laser had been built. For a milliwatt helium–neon laser the answer is a few thousandths of a hertz, a coherence length a quarter of the way to the Sun — and no real laser of that kind has ever come close, because mirrors move. For a semiconductor laser the answer is megahertz, and the measured lines were wider still, by a factor nobody expected.

 An attraction found by counting. How often two large discs are found at each gap between their surfaces, relative to how often they are found far apart, in a direct simulation of two discs among 270 small particles that pass through each other freely and do not enter the discs (dots), and the prediction from counting the small particles' available area (line); small particles a quarter of the large radius. Nothing in the simulation attracts anything: every move that does not put a small particle inside a large one is accepted. Yet the discs are found touching 2.7 times as often as apart (2.6 predicted), because touching returns the overlap of their exclusion zones to the small particles, and the probability of a separation is proportional to the number of arrangements the small particles have at it. Beyond a gap of twice the small radius the zones no longer overlap and the effect vanishes. Written as a potential, −kT times the logarithm of that ratio, it is an attraction about 1.0 kT deep. Thermodynamics

The attraction that is the small ones' room

Put large particles into a crowd of small ones that cannot get inside them, and the large ones begin to stick together — though nothing in the mixture attracts anything. Each large particle keeps the small ones' centres out of a thin shell around it. When two large particles touch, their shells overlap, and the overlap is handed back to the small particles as room to move in. More room is more arrangements, more arrangements is more entropy, and the large particles are pushed together by the pressure of the small ones on their outer sides. Fumio Oosawa and Sho Asakura worked it out in 1954. The force is now used to purify DNA, sort colloids onto patterned surfaces and explain why the inside of a cell behaves as though its proteins were stickier than they are.

 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.

 Every toss already decided. The outcome of a coin toss as a map of how it was launched: the upward speed in metres per second across, the spin in radians per second up, with the region where the coin shows the face it started with shaded and the region where it shows the other face left clear. A coin spinning about a diameter and caught at the height it left is in the air for 2v/g and turns 2vω/g radians, so the boundaries are the hyperbolae vω = nπg/2, one for every half-turn. Near the origin the bands are wide: a slow toss with little spin is easy to control, and a conjuror's "honest" toss lives there. Nothing in the map is random. Every point is a definite outcome, and the bands are smooth curves — there is no fractal boundary, no region where a tiny change has an unpredictable effect. What makes a toss a fair bet is only how narrow the bands become further out. Mechanics

The toss that is random without chaos

A coin toss is the textbook random event and the textbook deterministic one at the same time. Its flight is a few lines of Newtonian mechanics with no chaos anywhere: two tosses that start almost alike stay almost alike, drifting apart only in proportion to time. Its outcome is fixed the moment it leaves the thumb, by a map of launch speed and spin carved into bands of heads and tails. It is a fair bet because the bands, at a real toss, are only a few per cent wide and no hand repeats itself that well — and it is not quite fair, because a coin wobbles as it flips, and the wobble keeps its starting face up a little more than half the time: 50.8 per cent, predicted in 2007 and counted in 2023 over 350,757 tosses.

 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.

 A pulse that comes out of the average. The cross-correlation of the two stations' records against lag, in seconds: for single sources at 0°, 45° and 90° from the line through the stations, each a copy of the noise's own ten-second wavelet shifted to that source's arrival difference — 20.0 s, 14.1 s and 0 s — and the average over all 720 sources round the ring, scaled up 9-fold to share the axes. The single-source correlations are wherever their geometry puts them. The average has peaks at ±20 s, the time a wave takes to cross 60 km at 3 km/s, one for travel from A to B and one from B to A, and little between them. That pulse is the signal a seismometer at B would have recorded had an earthquake happened at A: the Green's function between the stations, recovered from noise that never came from either. Waves

The echo hidden in random noise

A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

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