A phase transition is a qualitative change of thermodynamic phase as control parameters such as temperature, pressure or chemical potential are varied. A genuine equilibrium singularity is a nonanalyticity of the appropriate thermodynamic potential in the thermodynamic limit; finite systems generally show rounded crossovers instead.
An order parameter is a macroscopic quantity that distinguishes the phases. It often vanishes in a symmetric disordered phase and becomes nonzero after spontaneous symmetry breaking, although density-driven transitions need not have that simple symmetry interpretation. It can be a scalar, vector or tensor. At a first-order phase transition an appropriate first derivative of the free energy is discontinuous, and the order parameter commonly jumps; at a continuous phase transition the order parameter can approach zero continuously while a susceptibility or a higher derivative becomes singular.
Three examples are:
- A ferromagnet below its Curie temperature has a nonzero spontaneous magnetization in zero applied magnetic field. Above the transition, the equilibrium magnetization vanishes in zero field. The direction of selects a broken-symmetry state.
- Near a liquid-gas critical point, the density contrast distinguishes the coexisting liquid and gas. It tends to zero at the critical point; below that point, changing the conjugate chemical potential through coexistence switches between low- and high-density phases.
- At an isotropic-to-nematic transition of a liquid crystal, the nematic order parameter is a symmetric traceless tensor, for example , where is a molecular orientation. It vanishes in the isotropic phase and is nonzero when the molecules preferentially align in a nematic liquid crystal.
These examples also illustrate why an order parameter must reflect the physical distinction between phases rather than merely be any quantity that changes with temperature.
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