What is Phases and order parameters?

A phase transition is a non-smooth change of a system’s equilibrium state as a control parameter ($T$, $P$, $B$) crosses a boundary. Its bookkeeper is the order parameter: a quantity zero in the disordered phase and nonzero in the ordered one — the magnetisation $m$ of a ferromagnet, the density difference $\rho_{\text{liq}} - \rho_{\text{gas}}$ of a fluid, the condensate fraction of Chapter 19. Transitions come in two kinds: first-order (order parameter jumps; latent heat; phases coexist — boiling, melting) and continuous (order parameter grows from zero; no latent heat; wild fluctuations — the Curie point, the critical point, superfluid helium). When the ordered state must choose among equivalent options — a magnet’s north has to point somewhere — the transition breaks a symmetry that the underlying laws respect: a theme that will return, at the highest stakes, with the Higgs mechanism. The engine behind every case is the free energy $F = E - TS$ (Proposition 17.2): energy favours order, entropy favours disorder, and $T$ sets the exchange rate — phase boundaries are where the ledger ties.