Phase transition (Gibbsian viewpoint)

Consider a lattice spin system with fixed interaction (e.g. via an interaction potential ) and parameters such as inverse temperature $\beta$ and external field (see external field coupling ). Let $\mathcal{G}(\gamma)$ be the set of infinite-volume Gibbs measures solving the DLR equation for the associated Gibbs specification $\gamma$.

A Gibbs phase transition is commonly signaled by non-uniqueness:

• there exist at least two distinct measures $\mu,\nu \in \mathcal{G}(\gamma)$.

Equivalently (in many standard settings), different boundary conditions produce different infinite-volume limits of finite-volume Gibbs measures .

A complementary thermodynamic signature is non-analyticity of the lattice pressure in the thermodynamic limit as a function of parameters. Both viewpoints are widely used; non-uniqueness corresponds most directly to phase coexistence, while non-analyticity captures both first-order and continuous critical phenomena.

Key properties

• Coexistence of pure phases: When non-uniqueness holds, $\mathcal{G}(\gamma)$ contains multiple pure phases , i.e. multiple extremal Gibbs measures distinguished by macroscopic observables (magnetization, density, etc.).

• Order parameter behavior: A phase transition is often accompanied by a qualitative change or discontinuity in an order parameter ; for ferromagnets, this can be spontaneous magnetization and spontaneous symmetry breaking .

• Long-range correlations near criticality: At or near a continuous transition, the correlation length may diverge and the susceptibility may become large, reflecting critical fluctuations even if the Gibbs measure is unique at the critical point.

• Mixtures vs pure states: In the coexistence region, symmetry-invariant equilibrium states can appear as mixtures of pure phases , which are Gibbs but not extremal.

Physical interpretation

A phase transition marks a change in the set and nature of equilibrium infinite-volume states: the system can support qualitatively different macroscopic behaviors under the same microscopic rules. In lattice spin systems, this includes transitions between disordered and ordered phases (e.g. paramagnet to ferromagnet), the onset of symmetry breaking, and critical points characterized by scale-free fluctuations and long-range correlations.