How fast order
grows out of chaos
Scramble a magnet completely, then drop it into the cold — how quickly do patches of agreement grow, and is there a law for it?
Every experiment before this one measured a system that had already settled — heat it, wait, ask what state it reached. M15 asks a different kind of question, and it is the lab's first non-equilibrium rung: not what state, but how fast.
Start by scrambling a sheet of magnets completely — a coin flip at every site, which is what infinite temperature means. Then drop the temperature to well below the freezing point and simply watch. Small patches of agreement appear immediately, everywhere. Then they start eating each other. A big patch absorbs a small one, the boundary between them straightens and shortens, and the characteristic patch size grows. It never stops growing; it just gets slower, because the bigger a patch is, the flatter its walls, and flat walls move sluggishly.
The theory here is called Allen–Cahn, and its prediction is clean: the typical patch size should grow as the square root of time. Double the time and patches get about 1.41× bigger. This run measures that exponent by tracking how far apart two sites have to be before they stop agreeing — the correlation length — and fitting how that distance grows on a log–log plot. Over about 2.6 decades of time the patch length climbs from 4 sites to 73, and the fit comes out at 0.486 against the predicted 0.5.
That it lands slightly below one half is expected, not a defect. Two-dimensional coarsening is known to approach its asymptotic law from below, and the run can show this directly: fit only the late portion of the data and the exponent drifts up to 0.496. An independent second estimator, built from the energy rather than the correlation length, gives 0.469. The defensible statement is that the exponent is somewhere around 0.47–0.49 and heading for one half — not that it is 0.486 to three decimals.
left · television static, then mottling, then black and white regions swallowing one another — visibly, relentlessly, and ever more slowly. The ember circle's radius is the currently measured domain length, so the number on the plot is visible as an actual length on the actual picture. The small panel in the corner is the wrong tool: the same lattice at the same temperature updated with Wolff cluster moves, which flip whole domains at once and order it almost instantly. It is the run's number-one documented failure mode — an algorithm that annihilates the very thing being measured · right · domain length against time on log–log axes, with a reference line of slope ½ anchored where the clean fit window begins.
The panel fits two windows and prints both, because on a lattice this size the window is the uncertainty. The clean window — domain lengths from 2 to 8 sites, before the lattice edge is anywhere near relevant — typically lands within a few hundredths of the published 0.47–0.49, on a quarter of the area and a single seed. The late window — 8 to 32 sites — comes out visibly steeper, and that is not better physics arriving. It is the lattice running out of room: once only a handful of domains remain, the correlation length starts reporting the box rather than the coarsening, and the slope runs away. Watching a well-behaved measurement turn into a bad one purely by extending its window is the most useful thing this small demo can show you, and it is the same failure mode the ±0.02 band on the published number is guarding against. The marked point is where the fit stops using the data.
The published run — 2026-07-07 — quenched a 512 × 512 lattice from infinite temperature to T = 1.498, which is 0.66 of the critical temperature, and followed it for 8 000 sweeps, averaging 48 independent random quenches. It finished on a GPU in 34 seconds. The correlation-length estimator fitted a slope of 0.486 with an R² of 0.9999 — and that R² is the trap. A statistical error of ±0.001 measures only how straight the log–log line is, not how right it is.