🧲 Ising Model — Phase Transition

In 1944, Lars Onsager derived a closed-form expression for the free energy and spontaneous magnetisation of the infinite 2D Ising model with no external field, proving rigorously that a genuine phase transition occurs at Tc = 2J/(kB·ln(1+√2)) ≈ 2.2692 J/kB. It remains one of the very few exactly solved interacting many-body models in statistical mechanics, and this simulation plots that exact curve directly against live Monte Carlo data so you can see the theory and simulation agree.

The Onsager solution is exact only in the thermodynamic limit of an infinite lattice. Any finite simulated lattice (even 200×200) shows finite-size effects: rounded transitions instead of a sharp kink, residual magnetisation slightly above zero above Tc, and statistical noise from the limited number of Monte Carlo sweeps. Increasing the Lattice size slider visibly sharpens the simulated curve toward the exact theoretical one.

A random spin is selected and the energy change ΔE from flipping it is computed using only its four nearest neighbours and the external field H. If ΔE ≤ 0 the flip is always accepted since it lowers total energy; if ΔE > 0 it is accepted with probability exp(−ΔE/kBT). Repeating this across the whole lattice many times (one "sweep" per N² attempts) drives the system toward thermal equilibrium at temperature T.

Susceptibility χ = N²(⟨m²⟩ − ⟨m⟩²)/kBT quantifies how strongly the system's magnetisation fluctuates. Exactly at Tc, spin correlations extend across the entire lattice, so even small thermal kicks produce large collective magnetisation swings, and χ diverges in the infinite-lattice limit as |T − Tc|^(−7/4). In a finite simulated lattice this shows up as a tall, finite peak rather than a true mathematical divergence.

A nonzero H energetically favours one spin orientation over the other, breaking the symmetry that makes the H = 0 transition sharp. With H ≠ 0, magnetisation varies smoothly with temperature and there is no true discontinuity in derivatives of the free energy — the sharp second-order transition exists only exactly along the H = 0 line, ending at the critical point (Tc, 0).