Illustrated Physics
physics, drawn
Concept

The second law — where it appears

The rule that the total entropy of an isolated system never decreases, which supplies physics with its only fundamental direction of time. It is a statement about overwhelming likelihood rather than impossibility, and its strength comes entirely from how large the numbers of arrangements are.

Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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

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

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

The ceiling on every engine, set before it was designed

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

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

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

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

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

The entropy that lives on a surface

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

thermodynamics · Entropy 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 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 most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

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

The bit that has to be paid for

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

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

The engine that has to finish

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

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

The glow that says nothing about the surface

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

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

The second law, with a probability attached

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

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

The engine a fluctuation cannot run

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

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

The area that is not allowed to shrink

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

astrophysics · Horizons Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.

The gas that flows towards more of itself

Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

thermodynamics · Diffusion Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K.

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

optics · Etendue The one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

The estimate that misses by a hundred and twenty

Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

astrophysics · Planck scale A fridge with no work going into it, and its ceiling. How much heat a three-reservoir machine can lift out of a cold space per unit of heat supplied to drive it, against the temperature of the driving heat, for 3 cold temperatures and an ambient of 300 kelvin. No work enters or leaves: the machine takes heat in at the top, takes heat in at the bottom, and rejects the sum at ambient. That it can do anything at all is the surprise — the second law allows heat to be moved up a gradient provided a larger flow is moved down one, and the accounting is a single inequality in the three entropy flows. The ceiling is the product of two familiar expressions, and at 450 kelvin driving a 253-kelvin space it is 1.79. The marks are what real machines achieve, which is a fifth to a third of it — absorption refrigeration is not efficient and is chosen when the heat is free and the silence and the absence of moving parts are worth something.

A fridge with no work going into it

Every engine here so far turns heat into work or work into a heat flow. A machine exchanging heat with three reservoirs and doing no work at all can still move heat from cold to hot, and the ceiling on how much is the product of two Carnot expressions — an engine's efficiency times a fridge's coefficient of performance. A gas flame makes ice, and the accounting is one inequality in three entropy flows.

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

The count that no observer can disagree about

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

relativity · Relativistic thermodynamics

Named alongside it

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

EntropyIrreversibilityReversibilityTemperatureBlackbodyThe Carnot cycleDetailed balanceEfficiencyEquilibriumFree energyMicrostatesWork

All concepts