Thought Toys · Emergence · Exhibit 43
A grid of tiny magnets, each one only caring which way its four neighbours point. Heat has no opinion about direction — yet at one precise temperature the whole sheet suddenly agrees, or suddenly forgets. Order appears from nothing, and vanishes just as sharply.
The lattice — 100×100 spins updown
Order parameter |m| versus temperature this runexact (Onsager)
T = 2.27 · at the knife-edge
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Every cell is a spin — amber for up, blue for down. It has no memory and no leader; each moment it only asks whether flipping would lower the energy of its four immediate neighbours. If so, it tends to flip. If flipping would raise the energy, it still flips sometimes — more often when it's hot. That single rule, run over and over, is the whole model.
At low temperature the neighbour-pull wins: spins fall into step and grow large amber or blue domains that swallow most of the grid, and the magnetization |m| — how lopsided the up/down vote is — sits near 1. Crank the temperature up and thermal flips overpower the pull; the grid dissolves into a fine, even speckle and |m| drops to 0. The remarkable part is how it drops: not gradually, but almost vertically, at one exact temperature. That's a phase transition — the same kind of sudden, collective switch as ice melting or a magnet losing its pull when heated red-hot.
Park the slider right at Tc and watch: the domains never settle: blotches of every size flicker in and out, the lattice can't decide. That critical shimmer is the system poised exactly between order and chaos.
improve/verify/43-ising.js): the page's own
Metropolis lattice, started aligned, holds |m| ≈ 0.98 at T = 1.5 but melts to ≈ 0.06 at T = 3.5 (a 0.9 drop
across Tc), and the ΔE shortcut matches a full-energy recompute exactly on every one of 400 test flips.
Negative control: strip out the Boltzmann bias — accept every proposed flip — and
the lattice never orders even at T = 1.5 (|m| ≈ 0.02), proving the order is manufactured by the
energy-lowering rule, not by the grid or the low temperature alone.
Also in Emergence: Phantom Traffic Jams →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 43.