🔥 Phase Transition Explorer – 2D Ising Model¶

This simulation demonstrates the most fundamental phenomenon in statistical physics: a phase transition between order and disorder.

🧠 Idea¶

A 2D lattice of magnetic spins (each +1 or −1) interacts with its nearest neighbours. The Hamiltonian is:

H = −J Σ s_i · s_j       (J = 1, sum over nearest neighbours)

At low temperature, spins align spontaneously (ferromagnetic order, |M| ≈ 1). At high temperature, thermal noise destroys correlations (disorder, |M| ≈ 0).

The transition happens sharply at the Onsager critical temperature:

T_c = 2 / ln(1 + √2) ≈ 2.269

Why this matters for the repository¶

Concept	Connection
Edge of Chaos	At T_c, correlations diverge — the system is maximally sensitive and complex
Repo Axiom 2	H(X) is neither 0 (frozen) nor maximal (noise) — life happens at the edge
Self-Organized Criticality	The Ising model shows why criticality is special; SOC shows how systems reach it
System Intelligence	Near T_c: P (prediction) is maximal, R (regulation) is fragile, A (adaptation) peaks

🎮 Controls¶

Key	Action
`←` / `→`	Decrease / increase temperature by 0.1
`r`	Reset the spin grid
`ESC`	Exit

📊 What You See¶

The display shows four panels:

Spin grid — Red/blue for ±1 spins. Watch domains form and dissolve.
Magnetisation |M| — Order parameter. Drops from ~1 to ~0 at T_c.
Energy per spin — Rises from ~−2 (ordered) towards 0 (disordered).
Phase diagram — Your current (T, |M|) overlaid on Onsager's exact solution.
Energy fluctuations — Proxy for specific heat C_v. Peaks at T_c.

▶ Run¶

cd simulation-models/phase-transition-explorer
python3 phase-transition-explorer.py

Experiment ideas¶

Start at T = 0.5 (deep order) and slowly sweep to T = 4.0 — watch the transition
Start above T_c and cool down — observe spontaneous symmetry breaking
Stay at T_c and watch scale-free fluctuations (large correlated domains appear and disappear)
Reset the grid at T_c and watch how long it takes to reach equilibrium (critical slowing down)