Generator

A path through a crowd

One function in the fluids library, called 26 times across 6 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 a path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

mean-free-path is one function in lib/figures/fluids.js — matter that will not hold a shape, and the forces in it. 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.

A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

A path through a crowd

The options are the ones How far a molecule gets passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Mean free path against how crowded it is

The options are the ones How far a molecule gets 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 mean free path against the number of scatterers per unit area, on logarithmic axes: a straight line of slope minus one, because doubling the crowd halves the distance between meetings. Air at room conditions sits far off the right of any drawable version of this — about 68 nanometres, some two hundred times a molecule's own size, which is the ratio that lets a gas be treated as a continuous fluid at all.

A hundred thousand years to cross seven hundred thousand kilometres

The options are the ones How far a molecule gets passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

How long energy takes to diffuse out of a body the size of the Sun, against the mean free path of the carrier, both logarithmic. The time is R² divided by λ and the speed of light, which is the random walk's square root read backwards, and the line has slope -1.0000 against an exact −1: shortening the free path by ten lengthens the journey by ten. At an opacity of 0.03 square metres per kilogram and a mean density of 1408 kilograms per cubic metre the free path is 2.4 centimetres and the escape takes 2.2e+3 years. The same distance in a straight line takes 2.3 seconds. That ratio — a factor of about 3e+10 — is the single most important number about the inside of a star, because it is why a star is opaque, why it has a temperature gradient rather than a temperature, and why nothing that happens in the core is visible from outside on any human timescale. The approximation here is a uniform sphere at the mean density, and it is worth naming: the real Sun is a hundred times denser at the centre than on average, and integrating the walk through that profile raises the answer to about 1.7e+5 years. Two orders of magnitude, all of it the density profile. The slope is untouched by it, and the slope is what the argument rests on.

Mean free path against how crowded it is

The options are the ones How far a molecule gets passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

A path through a crowd

The options are the ones How far a neutrino gets passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

What checks it

physicscheck asserts something about mean-free-path 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

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

Astrophysics

How far a neutrino gets

A mean free path is one over the number density times the cross-section, and nothing else. Change only the cross-section — by twenty-eight powers of ten — and the same arithmetic that gives a molecule seventy nanometres in air gives a neutrino a light-year of solid lead.

Thermodynamics

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

Astrophysics

The light that takes a hundred thousand years to leave

A neutrino made in the Sun's core is at the surface in two and a third seconds. A photon made beside it takes something like a hundred thousand years, through the same material, over the same seven hundred thousand kilometres — and the whole of the difference is one length, entering the answer squared.

Thermodynamics

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

Thermodynamics

The walk that comes home

A particle wandering at random on a line returns to where it started, with certainty. On a plane it returns, with certainty. In three dimensions the probability is 0.3405 — so two out of three molecules released in a room never pass through their starting point again, and the difference between the cases is not a matter of degree.

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