Critical exponent

Near a critical point, thermodynamic quantities do not vary smoothly; they follow power laws in the reduced temperature t = (T − T_c)/T_c. The exponents in those laws are the critical exponents, and each has a conventional name: the order parameter vanishes as (−t)^β, the susceptibility or compressibility diverges as |t|^(−γ), the specific heat as |t|^(−α), the correlation length as |t|^(−ν), and the critical isotherm follows m ~ h^(1/δ).

They are not independent. Scaling relations — Rushbrooke's α + 2β + γ = 2, Widom's γ = β(δ − 1), Fisher's γ = ν(2 − η), and the hyperscaling relation 2 − α = dν — leave only two of them free. That the relations hold at all was strong evidence, before anyone could explain it, that some rigid structure was generating the numbers; the renormalisation group later showed the structure was the flow of theories under coarse-graining, with the exponents as its eigenvalues.

The remarkable property is what the exponents do not depend on. Water, argon and xenon share one set; a 3D Ising ferromagnet and a demixing binary alloy share the same set again. Only the dimensionality of space and the symmetry of the order parameter matter — every microscopic detail drops out. Mean-field theory predicts β = 1/2 and is wrong; the measured value for three-dimensional systems is 0.326, and Onsager's exactly-solved 2D Ising model gives 1/8.

Guggenheim's 1945 collapse of eight fluid coexistence curves onto one line gave the first clear measurement of β ≈ 1/3 rather than the mean-field 1/2; Onsager's 1944 solution had already produced the first exact exponent, 1/8 in two dimensions. Wilson's renormalisation group explained where they come from in 1971–1974.