the Gibbs measure and the DLR equations
/ DLR = Dobrushin-Lanford-Ruelle; Dobrushin: do-BROO-shin; Ruelle: roo-ELL /
On a finite system the Gibbs (Boltzmann) measure is simple: weight each configuration by e^(-beta H) and normalize. But statistical mechanics lives in the infinite-volume (thermodynamic) limit, where the energy H is an infinite sum and the partition function diverges, so the naive formula is meaningless. The Dobrushin-Lanford-Ruelle (DLR) equations are the right definition of an infinite-volume Gibbs measure — and the framework in which a phase transition becomes, precisely, the existence of MORE THAN ONE such measure for the same interaction.
Instead of demanding a global Boltzmann formula, the DLR approach specifies all CONDITIONAL distributions consistently. Fix an interaction (a family of local energy terms). For each finite region Lambda this defines a finite-volume conditional measure: given the configuration outside Lambda (the boundary condition), the configuration inside Lambda has the Boltzmann distribution gamma_Lambda determined by the energies of bonds touching Lambda. A probability measure mu on the infinite configuration space is a Gibbs measure for the interaction if its conditional distribution on every finite Lambda, given the outside, equals exactly this specification gamma_Lambda — that is, mu is consistent with the family { gamma_Lambda }. These consistency relations are the DLR equations. The set of all such mu is a convex set; its extreme points are the pure phases, and a general Gibbs measure is a mixture of pure phases (an integral over the extreme points, by Choquet theory). Uniqueness of the Gibbs measure means the boundary condition is forgotten in the infinite-volume limit (a unique phase); NON-uniqueness — two or more Gibbs measures — is the rigorous definition of a first-order phase transition / coexistence of phases.
This abstraction is the modern language of equilibrium statistical mechanics, subsuming the Ising model (where the + and - states are two distinct Gibbs measures below beta_c, hence spontaneous symmetry breaking) and connecting to ergodic theory (extreme Gibbs measures are exactly the ones with trivial tail sigma-algebra). The honest subtleties are real: existence of at least one Gibbs measure is not automatic but holds under compactness/quasilocality conditions; uniqueness can be guaranteed by Dobrushin's uniqueness condition (a smallness criterion on the interaction) at high temperature, but FAILS at low temperature in d >= 2 — that failure IS the phase transition. And in d = 1 with finite-range interactions there is always a unique Gibbs measure, the rigorous reason one-dimensional systems do not exhibit phase transitions.
For the 2D Ising model below beta_c, take infinite-volume limits with all-plus and all-minus boundary conditions: they yield two DISTINCT Gibbs measures mu+ and mu- (with magnetizations +m* and -m*), both satisfying the same DLR equations. Their existence as different solutions is precisely the phase coexistence. Above beta_c both boundary conditions give the same unique Gibbs measure — no transition.
Non-uniqueness of DLR solutions = phase coexistence; the Ising + and - states below beta_c are the model case.
A phase transition is, rigorously, the existence of MORE THAN ONE Gibbs measure for one interaction — not a singularity of a single measure. In d = 1 with finite-range interactions the Gibbs measure is always unique, which is why genuine 1D phase transitions do not occur.