Critical Slowing Down — Relaxation & Autocorrelation

Critical slowing down: Near a continuous phase transition (T→T_c), the relaxation time of order-parameter fluctuations diverges as τ ~ ξ^z ~ |T-T_c|^{-νz}, where z is the dynamic critical exponent (z≈2 for local dynamics). For the 2D Ising model: T_c = 2J/ln(1+√2) ≈ 2.269 J. The autocorrelation time τ_int measures how long the system "remembers" its state — it grows powerfully near T_c. The autocorrelation function C(τ) = [⟨m(0)m(τ)⟩ − ⟨m⟩²] / [⟨m²⟩ − ⟨m⟩²] decays exponentially ~ e^{-t/τ} far from T_c, but as a power law at T_c. This limits Monte Carlo efficiency and motivates cluster algorithms (Wolff, Swendsen-Wang) that eliminate critical slowing down.