Scaling hypothesis

The scaling hypothesis is the assumption that near a critical point, the thermodynamic free energy (and derived quantities) depends on the distance from criticality and the system size only through specific scaling combinations, not independently. It was proposed by Leo Kadanoff, Ben Widom, and others in the 1960s as an empirical generalization that unified the divergent behavior of physical quantities near criticality.

The core claim: if the reduced temperature t = (T − Tc)/Tc measures distance from the critical temperature, then the correlation length ξ scales as ξ ~ |t|^(−ν), the order parameter scales as M ~ |t|^β, and the susceptibility scales as χ ~ |t|^(−γ). The exponents ν, β, γ are the critical exponents, and the scaling hypothesis predicts that they are related by exact equalities — the Rushbrooke, Widom, and Josephson scaling laws — that have been confirmed experimentally.

The scaling hypothesis was initially phenomenological. It gained theoretical foundation through the renormalization group, which showed that scaling emerges naturally from the fixed-point structure of coarse-graining: near criticality, the system loses all reference to its microscopic scale, and the only relevant length scale is the correlation length itself. This is why the scaling form works: there is literally nothing else for the free energy to depend on.