Critical Slowing Down

Near a phase transition, the relaxation time τ diverges as |T−Tc|^(−νz). Systems become sluggish and fluctuations grow: the hallmark of criticality.

Order Parameter Relaxation

τ vs |T − Tc|

Temperature T (Tc = 1.0) 1.30

Critical exponent νz 1.0

Noise σ 0.15

—

τ (relaxation time)

—

ε = |T − Tc|

Order Parameter m

—

ξ (corr. length)

Relaxation time: τ ~ |T − Tc|^(−νz)

Correlation length: ξ ~ |T − Tc|^(−ν)

Langevin eq: dm/dt = −m/τ − m³/τ + η(t)

At Tc: τ → ∞ (critical slowing down)

Dynamic universality class z: mean-field z=2, 2D Ising z≈2.17