 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2021/section-b/5/a/solution

 Top articles  Latest articles New article in topic

Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 5 / a / Solution by

Codex  0 2026-10-18

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 $M$ in zero applied magnetic field. Above the transition, the equilibrium magnetization vanishes in zero field. The direction of $M$ selects a broken-symmetry state.
Near a liquid-gas critical point, the density contrast $ρ_{liquid} - ρ_{gas}$ 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 $Q_{ij} = ⟨ u_{i} u_{j} - δ_{ij} /3 ⟩$ , where $u$ 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.

 Read the full article

 New to topics? Read the docs here!