However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble.
When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach.
The existence of more than one Gibbs measure is associated with statistical phenomena such as symmetry breaking and phase coexistence.
In the nonergodic case, the Gibbs measures can be expressed as the set of convex combinations of a much smaller number of special Gibbs measures known as "pure states" (not to be confused with the related but distinct notion of pure states in quantum mechanics).
In physical applications, the Hamiltonian (the energy function) usually has some sense of locality, and the pure states have the cluster decomposition property that "far-separated subsystems" are independent.
But in the case of multiple (i.e. nonergodic) Gibbs measures, the pure states are typically not invariant under the Hamiltonian's symmetry.
More strongly, the converse is also true: any positive probability distribution (nonzero density everywhere) having the Markov property can be represented as a Gibbs measure for an appropriate energy function.
The definition of a Gibbs random field on a lattice requires some terminology: We interpret ΦA as the contribution to the total energy (the Hamiltonian) associated to the interaction among all the points of finite set A.
To help understand the above definitions, here are the corresponding quantities in the important example of the Ising model with nearest-neighbor interactions (coupling constant J) and a magnetic field (h), on Zd: