Mean-Field Ising: Critical Slowing Down

Relaxation dynamics near the ferromagnetic phase transition — T → T_c diverges correlation time

Parameters

Temperature T / T_c: 1.20 Initial magnetization m₀: 0.80

Critical slowing down: Near T_c, relaxation time τ ∝ |T−T_c|⁻¹ diverges. The mean-field equation dm/dt = tanh(m/T) − m has a fixed point at m=0 for T>T_c. Linearizing gives τ = 1/(1/T − 1) → ∞ as T→1.

τ ≈ —

m(t) = —