an order parameter

How do you put a number on 'how ordered' a system is? When water freezes, iron magnetizes, or a metal turns superconducting, the system passes from a featureless, symmetric state to one with a definite structure. An order parameter is a quantity engineered to capture exactly this change: it is zero in the disordered (high-temperature) phase and becomes nonzero once order sets in. It is the single most useful bookkeeping device in the theory of phase transitions.

Precisely, an order parameter is a thermodynamic variable that vanishes in the symmetric phase and takes a nonzero value in the ordered, symmetry-broken phase. Familiar examples: the magnetization M for a ferromagnet (zero above the Curie temperature, nonzero below), the density difference between liquid and gas near the critical point, the complex condensate wavefunction (superfluid amplitude) for a superfluid or superconductor, and the staggered magnetization for an antiferromagnet. In a continuous (second-order) transition the order parameter grows continuously from zero as the temperature drops below T_c, typically as a power law: M is proportional to (T_c - T)^beta, where beta is a critical exponent. The order parameter can be a scalar, a vector, or a complex number, and its symmetry — how many components it has and how they transform — is what determines the transition's universality class.

The order parameter is the central object of Landau theory, which builds the free energy as a power series in it, and of the whole modern picture of symmetry breaking. Where you meet it: the magnet, the superfluid, the superconductor, the liquid crystal, even the Higgs field of particle physics. An honest caveat: the choice of order parameter is not unique and requires physical insight — you must identify what symmetry is broken; a poorly chosen order parameter can obscure the very transition you are trying to describe.