Critical exponents

Critical exponents are the dimensionless numbers that characterize the behavior of physical quantities near a phase transition. They describe how quantities such as magnetization, susceptibility, heat capacity, and correlation length diverge or vanish as the system approaches its critical point.

The standard notation uses Greek letters: β for the order parameter, γ for the susceptibility, α for the heat capacity, ν for the correlation length, and η for the correlation function decay. These exponents are not arbitrary. They are constrained by scaling relations such as the Rushbrooke equality (α + 2β + γ = 2) and the Widom scaling law (γ = β(δ − 1)), which follow from the assumption that the free energy near criticality is a generalized homogeneous function.

The profound fact about critical exponents is their universality: systems with completely different microscopic physics — a ferromagnet and a liquid-gas system — share the same exponents if they belong to the same universality class. This was confirmed experimentally before it was understood theoretically, and it remains one of the central puzzles that the renormalization group resolved.