STUDY #10 · 2026 · IN OBSERVATION

Ising Model

A model-driven visual study of a magnet's order-disorder phase transition.

MOVING IMAGE — TEMPERATURE ARC T 1.8 ⇄ 2.9 across Tc 2.269 · moke

WHAT IS THIS

A ferromagnet is made of countless atomic spins, each preferring to point the same way as its neighbours. Heat fights that alignment. At low temperature order wins and the material magnetises into large domains; at high temperature noise wins and the spins point every which way. The two are separated by a sharp critical temperature, Tc, where domains of every size coexist at once — scale-invariance, the lattice analogue of critical opalescence.

The Ising model is the minimal lattice version of this: spins s = ±1 that lower their energy by aligning, updated stochastically at a temperature by Metropolis Monte Carlo. This study runs the 2D model in real time on the GPU as a two-pass checkerboard update, and sweeps the single control — temperature — across the transition. Unlike the smooth reaction-diffusion studies before it, this one is stochastic: it flickers as it settles.

magnetic domains just below the critical point T 2.10 (Tc ≈ 2.269) · J 1 · h 0 · init hot · moke · seed 33

Motif Ising model / ferromagnetism / order-disorder transition / critical point / Monte Carlo / magnetic domains

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation A lattice of spins that each prefer to align with their neighbours is held at a temperature. One dial, T, moves it through a sharp transition: large frozen domains below the critical point, fine thermal static above, and at Tc self-similar clusters at every scale at once. The dynamics are stochastic — the pattern flickers as it settles.

Reference Ernst Ising, "Beitrag zur Theorie des Ferromagnetismus," Zeitschrift für Physik, vol.31, 253-258 (1925); Lars Onsager, "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition," Physical Review, vol.65, 117-149 (1944).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

DETERMINISTIC VS STOCHASTIC

Gray-Scott grows, Cahn-Hilliard sorts, Ising decides.

Studies #02 · #08 — continuous fields Study #10 — Ising

The field smooth concentrations, continuous in space and value discrete spins s = ±1 on a lattice

How it moves deterministic PDEs — the same start replays the same film stochastic Monte Carlo — it flickers, and never replays

The control feed & kill rates (#02) · the quench mix (#08) one dial — temperature, through a sharp critical point

What it does #02 grows structure · #08 sorts a mixture decides between order and disorder — at Tc, at every scale at once

PARAMETERS EXPLORED

param meaning effect on the image

T temperature — the strength of thermal noise (Tc ≈ 2.269 in units of J/kB) the one master dial: low T freezes large domains · ≈Tc gives scale-free critical clusters · high T is thermal static

J coupling — how strongly neighbours prefer to align (ferromagnetic > 0) sets the effective temperature T/J; raising it pushes the field toward order

h external magnetic field breaks the up/down symmetry — favours the gold (up) phase and drifts the domains to one side

init initial state — hot / cold / half hot → quench coarsens from static; cold → heating melts; half starts with a single domain wall

sweeps/frame Monte-Carlo sweeps per rendered frame how fast the field settles — and how hard it flickers on the way

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

A lattice of spins s = ±1. Alignment lowers the energy; temperature decides whether that preference wins.

E = -J\sum_{\langle ij\rangle} s_i s_j \;-\; h\sum_i s_i

The energy: each neighbouring pair lowers it by aligning; an external field h tilts the balance toward one spin direction.

P_{\text{flip}} = \min\!\big(1,\; e^{-\Delta E / T}\big), \qquad \Delta E = 2\,s_i\big(J\textstyle\sum_{nb} s_{nb} + h\big)

The Metropolis update — flips that lower the energy always happen; flips that raise it happen with a probability set by the temperature. This is where T enters.

m = \big(1 - \sinh^{-4}(2J/T)\big)^{1/8}, \qquad T_c = \frac{2J}{\ln(1+\sqrt{2})} \approx 2.269

Onsager's exact spontaneous magnetisation (1944) — one of the very few interacting systems solved in closed form, and the curve this engine was verified against.

Inspired by the Ising model — a visual interpretation, not an exact reproduction.

SELECTED STILLS — 3

symmetry broken by an external field — the gold phase wins T 2.15 · h +0.05 · |m| ≈ 0.87 · a few navy islands persist

lodestone palette — black steel and steel blue T 2.12 · h 0 · palette lodestone

PROCESS — PARAMETER SWEEPS

Same seed, same duration — only the temperature changes. Order → criticality → disorder in a single row.

The temperature ladder T = 1.4 · 1.8 · 2.1 · 2.27 (≈Tc) · 2.4 · 2.7 · 3.2

VERIFIED AGAINST 1944

Checked against an exact solution — to three decimals.

The 2D Ising model is one of the very few interacting systems ever solved exactly: Onsager's 1944 closed-form curve for the spontaneous magnetisation. Before any image was kept, this engine's order parameter |m|(T) was traced against that curve — maximum error 0.000 for T < Tc − 0.3; the rounding just above Tc is the expected finite-size effect.

That check is the series' method in miniature: the model has to be right before the picture is allowed to be beautiful.

order parameter |m|(T) — Monte-Carlo (dots, N 128) vs Onsager's exact solution (line) max error 0.000 for T < Tc − 0.3 · finite-size rounding above Tc

COLOUR = MAGNETIC DOMAINS

The Ising model was born as a model of ferromagnetism, and real magnetic domains are imaged by the magneto-optical Kerr effect: the two magnetisation directions read as two tones split by a bright domain wall. This study keeps that mapping — the down phase is a cool navy steel, the up phase a warm gold (aligned with the field), and the domain wall, where the smoothed magnetisation crosses zero, glows.

Near Tc the milky, scale-spanning texture echoes critical opalescence. The colours are an artistic mapping of the spin field s and its domain walls, not spectral measurements.

the same physics in the lodestone palette — a magnetite section T 2.12 · h 0 · palette lodestone · silver walls

Palette moke — hue = smoothed magnetisation m (navy ⇄ gold) · the bright cream seam = the domain wall where m crosses zero.

REFERENCES

Ernst Ising, "Beitrag zur Theorie des Ferromagnetismus," Zeitschrift für Physik, vol.31, 253-258 (1925).
Lars Onsager, "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition," Physical Review, vol.65, 117-149 (1944).
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller & E. Teller, "Equation of State Calculations by Fast Computing Machines," The Journal of Chemical Physics, vol.21, 1087-1092 (1953).
R. J. Glauber, "Time-Dependent Statistics of the Ising Model," Journal of Mathematical Physics, vol.4, 294-307 (1963).

INTERACTIVE STUDY

The whole subject of this study fits on one slider. Temperature is live — drag it across the Tc tick and back, and the field melts into static and refreezes in place, no reset needed. It is a deliberately simplified instrument, capped in resolution with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTISING · METROPOLIS · T LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.