Why does heat break up a magnet’s order?

Why would warmth undo order? Ising Domains tests the question on a small model: a 16 × 16 grid of tiles, each a two-way spin that points up or down. Every matching pair of neighbors lowers the energy, so the lowest-energy grid is one solid color. Temperature decides how often the grid pays to leave that state.

▶ Try it yourself in Ising Domains: drag Temperature from 1.5 to 3.5 and watch the odds of one flip change.

Predict

Each tile has four neighbors, and the grid’s edges wrap around. Flipping a tile whose four neighbors all agree with it breaks four matches and costs 8 units of energy, shown as +8 J. Before you change anything, predict: if you raise the temperature from 1.5 to 3.5, does that flip become a little more likely, or a lot more?

Test one change

The tool opens with every spin up and Temperature at 1.50. Press Pause, then Step 1 try a few times. Most tries land on a tile whose four neighbors all agree. The panel shows its flip cost of +8 J and an odds bar of 0.48%, which is exp(−8 / 1.5). The try’s random draw then drops onto that bar, and the flip passes only if the draw lands inside it. Press Play: a few lone tiles blink over and back, but the sheet stays almost one color.

Now drag Temperature to 3.50, or press Hot · 3.5, and change nothing else. The same +8 J bar rises to 10%, about twenty times higher. The +4 J bar, for a tile with one disagreeing neighbor, climbs from 6.9% to 32%. Within seconds the sheet breaks into speckle, and coral seams, each marking a mismatched pair, spread across the grid.

Why it happens

Each try follows the Metropolis rule. A flip that lowers or keeps the energy always happens. A flip that costs energy ΔE happens with probability exp(−ΔE / kT). Temperature sits in the denominator, so raising it softens the penalty on every costly flip at once. At 1.5, heat almost never pays for breaking four matches, and a stray mismatch is soon closed by its neighbors. At 3.5, mismatches open faster than neighbors can close them, so the order breaks apart.

Read one run carefully

The odds bars are exact numbers. The grid is one seeded run, so it fluctuates and need not change steadily; the Order |m| readout wanders most near 2.27. To compare temperatures fairly, restart several seeds at each setting and look at the spread, not a single frame. The tool reports no averages, and this guide claims none.

What this model leaves out

Temperature is in model units of J/k_B, not kelvin, and sweeps count tries, not seconds. The grid is a 16 × 16 toy with wrap-around edges, no external field, no chemistry, no vector spins and no real magnetic dynamics. Lars Onsager’s exact solution puts the order–disorder transition of an infinite square lattice near kT = 2.27. A grid this small has no sharp transition, and the model predicts no Curie temperature for any real material.

Next question

Set Temperature to 0 and restart from Random. Uphill flips never pass, so seams can only shrink or hold still. Does the grid always settle into one color, or can a stripe get stuck? Try a few seeds and see.

Sources: E. Ising, “Beitrag zur Theorie des Ferromagnetismus”; L. Onsager, “Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition”; N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of State Calculations by Fast Computing Machines”.