Illustrated Physics
physics, drawn
Concept

Phase transition — where it appears

The change of a substance from one state to another at a definite temperature and pressure, absorbing energy while the thermometer holds still. A first-order one has a latent heat and two coexisting phases; a continuous one has neither, and only its derivatives misbehave.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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

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

thermodynamics · Phase change Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do.

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

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

What a system actually minimises

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

thermodynamics · Entropy The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

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

The engine that pays back more than it takes

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

thermodynamics · Heat engines The height a 1 mK difference lifts helium. The head of liquid that a temperature difference of 1 millikelvin can support across a superleak — a plug fine enough that the normal fluid cannot pass and the superfluid can — against the temperature it is done at. Only the normal component carries entropy, so warming one side makes the superfluid flow toward the warm side until the pressure difference balances, and equilibrium is at ΔP = ρSΔT. The height that supports is SΔT/g, in which the density cancels exactly; computed both ways here the two agree to machine precision. The numbers are the striking part: 2.0 mm at 1.2 K, 4.6 mm at 1.4 K, 9.2 mm at 1.6 K, 19.4 mm at 1.8 K, 46.9 mm at 2 K, from a temperature step a thousand times smaller than anything a hand could feel. Aim a light at the warm side and the liquid does not merely rise but jets out of the tube, which is the fountain effect Allen and Jones found in 1938 and the most direct demonstration that helium II is two fluids rather than one. The entropies used are measured values; everything else on this chart is computed from them.

The fountain a lamp can drive

Below two degrees above absolute zero, liquid helium behaves as though it were two fluids occupying the same space — one carrying all the entropy and all the viscosity, the other carrying neither. It is not a metaphor and not a mixture. Shine a light on one side of a fine plug and the liquid jets out of the tube, because a temperature difference of a thousandth of a degree is a pressure of a hundred pascals.

fluids · Superfluidity The screening that survives to zero frequency. How far a magnetic field gets into copper against how fast it is changing, drawn beside the London depth of a superconductor with a carrier density of 6.0e+28 per cubic metre. The metal's curve falls as one over the square root of the frequency — measured on the drawn curve as -0.5000 against −½ — and it is a straight line on these axes with no bottom: at a hundred hertz the field reaches nine millimetres in, and at zero frequency it reaches all the way through, because a normal metal screens by dissipating and a steady field dissipates nothing. The superconductor's line is flat at 21.7 nanometres. The frequency does not appear in the expression for it, so there is nothing for it to depend on, and the screening is as complete at zero frequency as at any other. The two lines cross at 9.0e+12 hertz, in the far infrared, and above that the ordinary metal is actually the better screen — which is a useful corrective, because a superconductor's advantage is not that it screens harder but that it does not need the field to be changing. What the picture cannot show is where the flat line stops: above the energy gap the pairs break, the superconductor becomes an ordinary metal, and the flat line turns into a sloping one.

The field that is pushed out

A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor.

electromagnetism · Superconductivity A stiffness that does not go to zero — it falls off a cliff. The renormalised stiffness of a two-dimensional superfluid against temperature, obtained by integrating the flow of the bare stiffness and the vortex fugacity, with the bare stiffness drawn for comparison. Below the transition the vortices are bound in pairs, screen one another and leave a finite stiffness; above it they unbind and there is none. What is remarkable is the value at which it disappears: 0.6430, against 2/π = 0.6366. That number does not depend on the material, on the bare fugacity, on the core energy, or on anything else — every two-dimensional superfluid loses its stiffness at exactly two over π times its own transition temperature, and measurements on helium films of widely different thicknesses fall on that line. A transition at which a quantity jumps to zero from a value nobody can adjust is unlike anything the usual classification of transitions describes.

The transition with nothing to order

Every transition in this collection so far has an order parameter — a quantity that is zero on one side and not on the other. In two dimensions a continuous symmetry cannot break at any temperature above zero, so there is nothing for such a quantity to be, and by the usual reckoning there can be no transition. There is one anyway, and what changes at it is whether vortices are bound in pairs.

thermodynamics · Phase change The ring picks a whole number and pays for the difference. The kinetic energy of the screening current in a superconducting ring, against the flux applied from outside, in units of the flux quantum. Each parabola belongs to one winding number — the number of times the condensate's phase turns round the ring, which has to be an integer because the wavefunction has to come back to itself. The ring cannot hold an arbitrary flux, so it holds the nearest whole number of quanta and drives a current to make up the difference; the energy of that current goes as the square of the mismatch, which is what each parabola is. The lowest curve at each applied flux is the state the ring is in, and the winding number changes at exactly the half-integers — 0.50 and 1.50 and 2.50 here. So the measurable properties of the ring are periodic in the applied flux with a period of one quantum, which is 2.0678 × 10⁻¹⁵ webers and is about the flux the Earth's field puts through a square micrometre.

Two lengths, and which one is longer

A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.

electromagnetism · Superconductivity Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

thermodynamics · Phase change How much a block knows about the rest. The entanglement between a block of a one-dimensional chain and everything outside it, against how long the block is, for three states of the same number of particles. The straight line is a randomly chosen state, whose entanglement is the block's length times the logarithm of two — a volume law, and what almost every state in Hilbert space does. The flat curve is the ground state of a chain with a gap: it saturates, varying by less than a twentieth of a per cent from a block of eight to one of forty, because the boundary of a one-dimensional block is two points however long the block is. The middle curve is the ground state of a gapless chain, which grows as the logarithm of the size with a coefficient measured here as 0.333 against the third that conformal field theory gives. Both ground states are enormously less entangled than a random state, and that is not a detail about chains: it is why a ground state can be written down at all.

The corner of Hilbert space that is ever visited

Monogamy between three parties says how much of a correlation a pair may hold. Read across a boundary in a many-body system it says something much stronger: the entanglement between a region and the rest scales with the boundary rather than the volume, for the ground state of anything with local interactions. That is why such a state can be written down at all — and why almost every state in Hilbert space is one that nothing ever prepares.

quantum · Entanglement

Named alongside it

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

EntropyLatent heatTemperatureEquilibriumFree energyOrder parameterThe Boltzmann factorDensityMicrostatesPressureQuantisationThe second law

All concepts