DLR equation

Let $\gamma = (\gamma_\Lambda)_{\Lambda\Subset\mathbb{Z}^d}$ be a Gibbs specification on the configuration space $\Omega$, and let $\mu$ be a probability measure on $\Omega$.

The DLR equation (Dobrushin–Lanford–Ruelle) says that $\mu$ is an infinite-volume Gibbs measure for $\gamma$ if, for every finite region $\Lambda$ and every bounded measurable function $f:\Omega\to\mathbb{R}$,

Equivalently, in terms of conditional expectations (see conditional expectation ), $\mu$ satisfies that its conditional distribution in $\Lambda$ given the outside sigma-algebra coincides with $\gamma_\Lambda(\cdot\mid\eta)$ for $\mu$-almost every exterior configuration $\eta$.

A measure $\mu$ satisfying the DLR equation is precisely a infinite-volume Gibbs measure .

Key properties

• Convex set of solutions. The collection of all DLR solutions for a fixed specification is convex: mixtures of solutions are solutions. This leads to the mixture decomposition viewpoint.

• Extremal measures and pure phases. Extreme points of the DLR solution set are extremal Gibbs measures , which correspond to pure phases in the usual physical interpretation.

• Phase transitions as non-uniqueness. A common mathematical formulation of a phase transition is the existence of more than one DLR solution (at the same parameters), often accompanied by phenomena such as spontaneous magnetization or spontaneous symmetry breaking .

• Local equilibrium principle. The DLR equation enforces that every finite window of the infinite system, conditioned on its exterior, is distributed as the corresponding finite-volume Gibbs law prescribed by the specification.

Physical interpretation

The DLR equation is the rigorous form of the Gibbs postulate for infinite systems: even without a global finite-volume normalization, the system is in equilibrium if every finite region looks like a Gibbs-distributed subsystem when conditioned on its surroundings. Non-uniqueness of DLR solutions captures coexistence of macroscopic phases.