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Concept

Diffusion

The spreading of a quantity from where there is more of it to where there is less, at a rate set by how sharply it varies.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

050100150200250300350400-60-40-20204060steps takendistance from the start√n

A walk that always comes home, until it does not

Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.

probability · random walk −ππ-11position around the ringtemperatureat the startt = 0.005t = 0.05t = 0.4a square profile on a ring, left to spread: harmonic m fades by exp(−m²t), so the 41th term is gone 1,681 times fasterthan the firstthe corners disappear immediately and the shape that remains is a single sine — which is why running the flowbackwards is hopeless: the information in the corners has been divided by a number this large

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

analysis · fourier series

Named alongside it

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

Binomial distributionConvergenceEigenfunctionExponential decayFourier analysisHeat equationIndependenceIrreversibilityLimitNormal distributionParityRandom walk

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