A square grid of spins, each up or down, is the simplest model of a magnet. This sim lets you set the temperature, watch the grid rearrange under one local rule, and measure how aligned the whole grid is.

What's on screen

• Spin up (+1): blue squares in the lattice. The title line above it gives L and T.
• Spin down (−1): amber squares.
• M trace (teal): the signed mean spin of the whole grid over the last 15 s, from −1 (all down) to +1 (all up).
• ⟨|M|⟩ plot (orange dots): one dot per ⏺ Record, with a bar for the sweep-to-sweep spread. Filled dots were measured on the current grid size, hollow ones on another.
• Exact curve and T_c lines (amber): Onsager's result for an infinite lattice, and where your grid's rise is expected. The Guided Inquiry reveals them after you have mapped the rise yourself.

Press ⌖ on a row to see where that control sits in the sim. In the Guided Inquiry some controls appear only at later cards; in Free Exploration they are all there.

Fixed: J = k_B = 1, zero external field, periodic edges.

A neighbouring pair contributes −J to the energy if its two spins match and +J if they differ. Each step picks one spin at random and proposes to flip it. The Metropolis rule accepts the flip with this probability:

• A flip that lowers or keeps the energy is always accepted.
• An uphill flip is accepted with probability e^−ΔE/k_BT. For ΔE = +4J that is 0.082 at T = 1.6 and 0.264 at T = 3.0, so heat is what buys disagreement.
• One sweep is L² attempts: Monte Carlo time, not seconds.
• This rule satisfies detailed balance, so a long run samples the equilibrium distribution at temperature T.

Below one temperature, matching neighbours win often enough that agreement spreads across the whole grid. Onsager's exact solution for an infinite lattice gives that temperature and the size of the order:

• M(T) is 0.98 at T = 1.6, and 0.91 at T = 2.0.
• An eighth root rises steeply: 1% below T_c it is already 0.69, and 0.1% below it is still 0.52.
• Cool gradually. A sudden quench can freeze the grid into two ordered stripes with domain walls, and the sim flags trapped stripes. Reheat and cool again.

With no external field, flipping every spin leaves the energy and every flip probability unchanged. Nothing in the rule prefers up, so the colour that takes over is decided by each run's random fluctuations, and both are equally likely.

• On a finite grid at zero field, the long-run average of the signed M is exactly zero: given enough time, a grid below T_c flips between its two ordered states.
• That waiting time grows exponentially with L. Well below T_c it is far longer than any session, so the trace stays at one sign.
• That is why the plot averages |M|. The signed average would cancel to zero and hide the order, while |M| counts up-order and down-order alike.

Close to T_c, ordered patches of every size coexist, and the grid needs very many sweeps before it forgets its present state. The signed M wanders slowly over a wide range instead of jittering about zero.

• Successive sweeps stay correlated, so 400 sweeps hold far fewer independent samples than they do at T = 1.6 or 3.0. Between 2.15 and 2.45, Record therefore takes three times as many: 600 to settle and 1200 to measure.
• The scatter between runs is large here. Repeated 64 × 64 records at T = 2.30, simulated with the sim's own rule, landed anywhere from about 0.1 to 0.6, while at T = 1.6 or 3.0 they repeat to within a few hundredths.
• That scatter is physics, not noise to be averaged away. The ± in the Record message is the sweep-to-sweep spread, not an error bar.

Onsager's curve is for an infinite lattice. An L × L grid cannot show a sharp drop, so its rise is rounded and sits slightly warm of T_c, and some ⟨|M|⟩ remains above it.

Exact T_c = 2.269. The last column is an average of repeated records; a single record scatters by about a third.

• The tail above T_c falls roughly as 1/L. It is shown, never subtracted.
• A larger grid gets closer to the exact curve, but near T_c it also needs longer runs.

• One idea, many materials. The order parameter changes name, not role: magnetisation in a magnet, the liquid–vapour density difference in a fluid. Each vanishes at its critical point, and systems that share a dimension and a symmetry share exponents. This is called universality.
• Real fluids and layered magnets. CO₂ has its liquid–vapour critical point at 304 K and 73.8 bar, in the three-dimensional Ising class (β ≈ 0.326, not this grid's 1/8). Layered magnets such as Rb₂CoF₄ behave as two-dimensional Ising systems, with a measured β close to 1/8.
• Simulation practice. Metropolis sampling is a standard tool in statistical physics. The slow decisions and finite-size shifts you measured here are the everyday obstacles near any critical point.

Key points

• A flip costs ΔE = 2J s_i Σ s_j. Downhill is always taken, uphill with probability e^−ΔE/k_BT.
• Below T_c, ⟨|M|⟩ rises steeply as (1 − T/T_c)^1/8.
• Near T_c the grid decides slowly, so records scatter.
• A finite L rounds the transition, shifts it warm by about T_c·0.8/L, and leaves a tail that falls as 1/L.

Common mix-ups

• "Order needs something that favours up." The field is zero and the rule is symmetric. Below T_c, agreeing wins in either direction.
• "It should settle half up, half down." A wall of mismatched neighbours costs energy. A hasty quench can still freeze into stripes.
• "The transition on screen is sharp, and ⟨|M|⟩ is zero above it." On a finite grid the rise is rounded, and a tail remains above T_c, about 0.04 on 64 × 64 at T = 3.0.

Things to try on your own

• Cool slowly from 3.0 and Record at six or more temperatures, then compare your dots with the exact curve.
• Record three times at T = 2.30, then three times at T = 1.6, and compare the spreads.
• Record at T = 2.5 on L = 32, 64 and 96 and compare how large the tail above T_c is on each.