Generator

Ways to arrange 10 coins

One function in the thermal library, called 45 times across 10 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws 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.

microstates is one function in lib/figures/thermal.js — cycles, distributions and the statistics underneath them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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.

Ways to arrange 10 coins

The options are the ones Entropy is a count, and the arrow of time is arithmetic passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

Ways to arrange 4 coins

The number of distinct arrangements giving each number of heads, for 4 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Ways to arrange 20 coins

The number of distinct arrangements giving each number of heads, for 20 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Ways to arrange 20 coins

The number of distinct arrangements giving each number of heads, for 20 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Whether the entropy doubles when the gas does

Entropy per particle against the number of particles, at fixed density and temperature, counted two ways. The upper line counts arrangements as though every particle carried a label, so that swapping two of them gives a different arrangement; its entropy per particle grows without limit as the sample grows, which no thermodynamic quantity may do — two identical flasks joined together would then have more than twice the entropy of one, and opening a tap between them would produce entropy from nothing. The lower line divides the count by the number of permutations of the particles, and its entropy per particle is flat to 0.0310 across a factor of 40 in size, and what is left of the drift is the leading Stirling correction, ln(2πN)/2N, which is falling toward nothing as the sample grows and is already invisible at any number of particles a flask contains. That division is the whole repair, it is worth exactly one factorial, and it says something physical: two arrangements differing only by which particle is where are not two arrangements. Nothing in classical mechanics requires that, and it had to be put in by hand for forty years before quantum mechanics said why.

What checks it

physicscheck asserts something about microstates that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Thermodynamics

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

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

Thermodynamics

Mixing what is already mixed

Let two different gases into each other's halves of a box and the entropy rises by a fixed amount that contains nothing about either gas. Do it with the same gas on both sides and it rises by nothing. Make the gases more and more alike and the answer does not converge — it jumps.

Thermodynamics

The bit that has to be paid for

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

Thermodynamics

The entropy that is still there at zero

The third law says a perfect crystal has no entropy at absolute zero. Ice has 3.41 joules per kelvin per mole left over, and the number can be recovered from one line of counting — two hydrogens near each oxygen and two far, six legal arrangements out of sixteen, R ln(3/2). The law has an escape clause and the escape clause is measurable.

Thermodynamics

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

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

Thermodynamics

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

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

The summer that reaches the cellar in December

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

Thermodynamics

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

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.

The whole library · All essays