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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. It sends a quantity's spread up like the square root of time rather than in proportion to it, which is why it is so bad at covering distance.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit.

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 A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

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 One walk at three magnifications, and the shape it is heading for. The same random walk over three windows, each ten times longer than the last and scaled vertically by the square root of ten, so all three look alike. Beside them, the exact distribution of the position after a few step counts, standardised, closing on the bell curve.

The walk that becomes a curve

Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.

probability · Random walk

Named alongside it

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

LimitNormal distributionRandom walkBinomial distributionContinuityConvergenceEigenfunctionExponential decayFourier analysisHeat equationIndependenceIrreversibility

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