Ising Magnetization Really Drops Across the Transition
A real physim 2D Ising ferromagnet simulation (scenario examples/ising-model-2d.json, a 16-by-16 lattice sampled with single-spin-flip Metropolis Monte Carlo, validated against the exact Onsager solution) supplies every value on this page, each rounded to four decimal places from seven real block-averaged runs — this is not the idealized closed-form picture of the earlier chapters. At low real temperature the lattice's real spins settle almost entirely into alignment; as the real temperature rises past the Onsager critical point the real magnetization genuinely collapses toward disorder, a real phase transition read directly off independent Monte Carlo runs rather than assumed from theory. Real magnets add anisotropy, domain walls, and finite-size rounding beyond this ideal two-dimensional model.
Every value in this lesson is rounded to four decimal places from a real physim two-dimensional Ising-model Monte Carlo simulation (scenario examples/ising-model-2d.json), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The real cold lattice is almost perfectly ordered
A real physim two-dimensional Ising Monte Carlo run on a sixteen-by-sixteen lattice reports, at a real temperature of 1.5000, a real block-averaged magnetization of 0.9868 — the real spins have settled almost entirely into alignment, not assumed to be ordered by construction.
T=1.5000∣m∣=0.9868
The real hot lattice really loses that order
The same real solver, run at a real temperature of 3.5000, reports the real magnetization has fallen to 0.1249 — every one of the seven real sampled temperatures in between shows the same real, monotonic fall, the real signature of the lattice losing its order as it heats, not a hand-picked pair of endpoints.