= Solution
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> $\mathbf M$ in zero applied <magnetic field>. Above the transition, the equilibrium <magnetization> vanishes in zero field. The direction of $\mathbf M$ selects a broken-symmetry state.
* Near a <liquid-gas critical point>, the density contrast $\rho_{\rm liquid}-\rho_{\rm gas}$ 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 $Q_{ij}=\langle u_i u_j-\delta_{ij}/3\rangle$, where $\mathbf u$ 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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