a critical exponent

Approach a critical point — the exact temperature and pressure where liquid and gas become indistinguishable, or the Curie point where a magnet loses its magnetism — and physical quantities start to diverge or vanish. Remarkably, they do so as clean power laws, and the exponents of those power laws are among the most precisely characterized numbers in physics. A critical exponent quantifies how sharply a quantity blows up or dies as you near the critical point.

Define the reduced temperature t = (T - T_c)/T_c, which measures the distance from criticality. Near t = 0 the key quantities follow power laws whose exponents are given standard Greek letters: the order parameter vanishes as M is proportional to |t|^beta (for T below T_c); the susceptibility diverges as chi is proportional to |t|^(-gamma); the specific heat behaves as C is proportional to |t|^(-alpha); the correlation length diverges as xi is proportional to |t|^(-nu); and exactly at T_c the order parameter responds to a field as M is proportional to h^(1/delta). These exponents are not independent — they obey scaling relations such as Rushbrooke's identity alpha + 2 beta + gamma = 2 and Fisher's relation gamma = nu(2 - eta), which follow from the scaling hypothesis and are confirmed experimentally.

Critical exponents matter because they are UNIVERSAL: wildly different systems sharing the same dimensionality and order-parameter symmetry have identical exponents, a fact explained by the renormalization group. An essential honesty: mean-field (Landau) theory predicts the 'classical' values beta = 1/2, gamma = 1, alpha = 0, delta = 3 — and these are simply WRONG in three dimensions, where the measured 3D Ising values are beta about 0.33, gamma about 1.24. Mean-field theory fails because it ignores the large fluctuations that dominate near a critical point below the upper critical dimension (four for the Ising class).