Critical Phenomena

Physics: Critical exponents describe power-law singularities: order parameter m ~ |T−T_c|^β, correlation length ξ ~ |T−T_c|^(−ν), susceptibility χ ~ |T−T_c|^(−γ), heat capacity C ~ |T−T_c|^(−α). 2D Ising exact (Onsager 1944): β=1/8, γ=7/4, ν=1, η=1/4; 3D Ising: β≈0.326, ν≈0.630 (no exact solution). Universality: exponents depend only on symmetry group and dimension, not microscopic details. Hyperscaling relation: 2−α = ν·d. The renormalization group (Wilson 1971, Nobel 1982) explains why systems with different interactions share identical critical exponents.