Mixtures of Gibbs measures

Let $\gamma$ be a Gibbs specification and let $\mu_1,\mu_2 \in \mathcal{G}(\gamma)$ be infinite-volume Gibbs measures .

A mixture of Gibbs measures is any convex combination

or more generally an integral (barycenter) over Gibbs measures:

where $\rho$ is a probability measure on the space of Gibbs measures.

Because the DLR consistency is linear in $\mu$, any such mixture is again a Gibbs measure in $\mathcal{G}(\gamma)$.

Key properties

• Convexity of the Gibbs set: The set $\mathcal{G}(\gamma)$ is convex, so mixing preserves the Gibbs property.

• Extremal decomposition: Any Gibbs measure can be decomposed into extremal Gibbs measures (the extreme points of $\mathcal{G}(\gamma)$). In this sense, mixtures are the generic non-extremal equilibrium states.

• Nontrivial tail behavior: Mixtures typically have a nontrivial tail σ-algebra: there can be tail events with probabilities strictly between $0$ and $1$, reflecting the random choice among phases “at infinity.”

• Symmetry-invariant but non-extremal states: In symmetric models, there are translation-invariant Gibbs measures that are mixtures of symmetry-broken pure states (see pure phase ). These measures can be invariant under a symmetry even when no extremal Gibbs measure has that symmetry.

Physical interpretation

A mixture Gibbs measure describes an ensemble that randomly selects among competing equilibrium phases. For example, in a low-temperature ferromagnet, a symmetry-preserving infinite-volume state can arise as a 50–50 mixture of the two magnetized phases. Such a mixture is a valid equilibrium state mathematically, but it represents phase coexistence (or uncertainty about which pure phase is realized) rather than a single homogeneous macroscopic phase in a typical large sample.