Illustrated Physics
physics, drawn
Concept

Relaxation — where it appears

The return of a disturbed system toward equilibrium, characterised by the time it takes rather than by any force. Comparing that time with the duration of an observation is what decides whether a material behaves as a solid or a liquid, and the same substance answers differently on different clocks.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

The same steady pull, twice. Spring force divided by the normal load, against time, with the far end drawn away at 100 µm/s in both traces and every property of the contact the same. The soft holder at 0.4 k_c produces events every 3.75 s, each of which reaches 0.94 m/s — 9.4e+3 times the speed it is being pulled at. The stiff one at 1.6 k_c settles to a straight line and stays on it.

The chatter a stiffer holder removes

A brake squeals, a bow sounds a string, a fault slips in jerks. The usual explanation is that static friction exceeds kinetic friction — and that explanation, taken seriously, predicts the jerking would happen no matter how the thing were held. It does not, and what decides is a length nobody mentions.

mechanics · Friction 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.

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.

fluids · Rheology An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

waves · Attenuation 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.

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.

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

The energy that refuses to be shared

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

thermodynamics · Equipartition 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.

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.

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

The entropy that depends on how fast it was cooled

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

thermodynamics · Third law The 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.

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.

electromagnetism · Magnetisation The condition three modes never quite satisfy. By how much three modes of a thirty-two mass chain fail to be in resonance — the sum of two mode frequencies minus the frequency of their sum, on a logarithmic scale, against the second of the two modes for four choices of the first. The leading nonlinear term couples modes in threes and the exchange accumulates only where this quantity is zero. It never is: the chain's dispersion is a sine, a sine is concave, and the sum of two of its values always exceeds the value at their sum. The smallest mismatch anywhere on the chain is at the two lowest modes and equals 2.156e-4 — which the scan finds and which is the cube of pi over four times the cube of one more than the mode count, checked here on chains from eight masses to two hundred and fifty-six. That closed form is the whole of why this is a finite-chain problem: the mismatch falls as the cube of the length, so a long enough chain is arbitrarily close to resonant and the continuum limit is exactly resonant, which is where the solitary waves come from.

The condition three modes never meet

Whether two modes of a chain can hand energy to a third is arithmetic on the dispersion relation, and for a chain of masses the answer is never: a sine is concave, so the sum of two frequencies always exceeds the frequency of their sum. The smallest shortfall anywhere on a chain of N masses is π³/4(N+1)³ — never zero, and never far from it — and a shortfall turns a transfer into a beat.

thermodynamics · Equipartition A living cell and a dead one, told apart by a frequency. The real part of the dielectrophoretic factor for a living cell and a dead one, ten micrometres across, in a liquid of conductivity 0.01 S/m, against frequency. The cell is a conducting interior inside a membrane five nanometres thick. The living cell's intact membrane blocks slow fields, so at low frequency it behaves as an insulator and is pushed out of strong field; above 42 kHz the membrane is short-circuited by its own capacitance and the conducting interior is felt, so the cell is pulled in; above 140 MHz the permittivities take over and it is pushed again. The dead cell's membrane leaks, its interior has lost ions, and it changes sign only at 8.8 MHz. In 2 bands — below 42 kHz and from 8.9 MHz to 140 MHz — the same field pushes the two in opposite directions.

The force whose sign a frequency chooses

A neutral particle in a non-uniform field is pulled towards strong field if it polarises more than the liquid around it and pushed away if it polarises less. With conduction in the picture, which of the two it does is decided by how fast the field alternates — and a living cell, a conductor wrapped in an insulator five nanometres thick, changes sign twice, where a dead one changes once. The same electrodes can then send the living cells one way and the dead ones the other.

electromagnetism · Dielectrics The thickness fixes the area, and the design only moves it. How much a metal-backed resistive sheet 7.5 mm from the metal absorbs, measured as ln(1/|Γ|), against wavelength, for sheets of 150, 250, 377, 800 Ω per square. Rozanov's bound says the area under any such curve, over all wavelengths, can be at most 2π² times the thickness: 147.9 mm here. The areas computed are 147.9 mm for 150 Ω, 147.9 mm for 250 Ω, 147.9 mm for 377 Ω, 69.7 mm for 800 Ω. Every sheet up to the impedance of free space meets the bound exactly and only moves its area between wavelengths; the more resistive sheet wastes part of it.

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

waves · Impedance 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.

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.

thermodynamics · Entropy A number of details that frequency cannot change. The number of separately resolvable depths along one line of an ultrasound image, the deepest echo divided by the axial resolution, against frequency, for dynamic ranges of 50, 60 and 80 dB in soft tissue; dashed, the same for a medium that absorbed like water, as the square of the frequency, with 60 dB. In tissue the count is D/(α₀Nc), the same at every frequency: 325 for 50 dB, 390 for 60 dB, 519 for 80 dB. It is a property of the tissue, through α₀, and of the electronics, through D, and changing the frequency only trades depth for sharpness one for one. In a medium with an f² law the count would fall as 1/f, from 88,548 at 1 MHz to 1,771 at 50 MHz. It is the near-linear absorption of tissue that makes the trade exact.

The depth an ultrasound image pays for its sharpness

An ultrasound scanner sees finer detail at higher frequencies, because shorter waves make shorter pulses, and sees less deep, because tissue absorbs higher frequencies more strongly. Both effects scale as one over the frequency, so they cancel in the one number that matters: how many separate depths fit along a line of the image. For soft tissue, which absorbs nearly in proportion to frequency, that number is about four hundred whatever the frequency — set by the tissue and by the scanner's electronics and by nothing else. A scan of the abdomen at three megahertz and a scan of the skin at fifty are the same picture at different magnifications.

waves · Attenuation

Named alongside it

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

DissipationErgodicityDampingEquilibriumEquipartitionHysteresisIntegrabilityMetastabilityNonlinearityNormal modeNumerical experimentResonance

All concepts