Critical Exponents & Scaling Collapse

Critical exponents: Near a second-order phase transition at T_c, physical quantities diverge or vanish as power laws of the reduced temperature t = (T-T_c)/T_c:
m ~ |t|^β (below T_c), χ ~ |t|^(-γ), ξ ~ |t|^(-ν), C_V ~ |t|^(-α)

Scaling hypothesis (Widom 1965): The free energy is a generalized homogeneous function: f(t,h) = |t|^(2-α) Φ(h/|t|^(βδ)). All exponents follow from just two independent ones (e.g., ν and η). This gives scaling relations: γ = ν(2-η), α+2β+γ=2 (Rushbrooke), γ=β(δ-1).

Universality: Systems with the same dimensionality d and order parameter symmetry n share identical exponents (universality class). Ising 3D (β=0.326, γ=1.237, ν=0.630) is exact for magnets, liquid-gas, polymer collapse — all sharing d=3, n=1.

Data collapse: Plotting m·|t|^(-β) vs h·|t|^(-βδ) for different temperatures should collapse onto a single universal curve if the exponents are correct.