Pure phase

Informally, a pure phase is a single, homogeneous thermodynamic state (no macroscopic coexistence of distinct phases). In the Gibbs/DLR framework for lattice systems, a standard rigorous identification is:

A pure phase at fixed interaction and parameters is an infinite-volume Gibbs measure that is extremal in the Gibbs set and (in many applications) invariant and ergodic under lattice translations when the interaction is translation-invariant .

This definition separates “indecomposable equilibrium states” (extremality) from mixtures (see mixtures of Gibbs measures ).

Key properties

• No decomposition into other phases: A pure phase cannot be expressed as a nontrivial convex mixture of other Gibbs states.

• Sharp macroscopic observables: Order parameters (see order parameter ) typically take a definite value within a pure phase; fluctuations average out in large volumes (“self-averaging”).

• Clustering inside a phase (typical): Many pure phases satisfy a clustering/mixing property: connected two-point correlations decay with distance (away from criticality). This distinguishes a single homogeneous phase from a random mixture.

• Phase multiplicity and transitions: The existence of more than one pure phase at the same parameters is a hallmark of a phase transition .

• Symmetry breaking: In models with global symmetries, distinct pure phases may be related by the symmetry and exhibit spontaneous symmetry breaking . A classic diagnostic is nonzero spontaneous magnetization in a ferromagnet.

Physical interpretation

A pure phase is what one expects to observe in a large, well-equilibrated sample when boundary conditions or preparation select a definite macroscopic state. Different pure phases correspond to different stable macroscopic behaviors (e.g. magnetized “up” vs magnetized “down”), while a mixture Gibbs measure represents an ensemble that does not select a single phase and thus encodes phase coexistence or random phase selection.