🧲 Ising Model
The Ising model (Ernst Ising, 1925) is the canonical model of statistical mechanics. A lattice of spins (±1) interact via nearest-neighbour coupling J; each spin aligns or anti-aligns with its neighbours and an external field H. The Metropolis–Hastings algorithm samples the Boltzmann distribution: a spin-flip is always accepted if ΔE ≤ 0, otherwise with probability e^(−ΔE/kT). Below the Curie temperature T_c ≈ 2.27 J/k spontaneous magnetisation appears: a second-order phase transition. Watch large spin domains form and melt as you sweep temperature. 🇺🇦 Українська
Parameters
Lattice
Physics of the Ising Model
The Hamiltonian is H = −J Σ sᵢsⱼ − h Σ sᵢ (sum over nearest-neighbour pairs). The exact 2D critical temperature is k_BT_c = 2J/ln(1+√2) ≈ 2.269 J. At T_c the correlation length diverges (ξ → ∞), giving scale-free spin clusters and critical slowing-down. The plot tracks magnetisation |⟨m⟩| over time; susceptibility χ = N(⟨m²⟩ − ⟨m⟩²)/kT peaks sharply at T_c.