The DLR equations and limit of local Gibbs states I  am interested in   mathematical theory  of equilibrium states at positive temperature. According to the monograph by Barry Simon The Statistical Mechanics of Lattice Gases'' at page 246,   the set of equilibrium states characterized by the DLR equations contains all limits of finite-volume Gibbs states associated to  the given interaction.    My question is about the converse, namely, 
Are  there equilibrium states satisfying the DLR equation which are however not written as a weak limit of finite-volume Gibbs states? I suspect that such an equilibrium state seems  exceptional (if it exists). I would like to know thorough model independent argument on the meaning ofthe smallest one'' that Simon's book indicates.      
 A: Yes, in general there are solutions of the DLR equations that are not limits of finite-volume Gibbs states. Now, let me make some additional comments on this matter.


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*An example of a Gibbs measure that cannot be obtained as the limit of finite-volume Gibbs measure can be found in Example 6.64 of our book (actually, Section 6.8.2 is explicitly devoted to the question you raised); this example is for the three-dimensional Ising model and was found independently by Miyamoto and by Coquille (the precise references are given in the book). By the way, I encourage you to have a look a Chapter 6 of our book, as I believe that the latter provides an easier introduction to this topic than Simon's one.

*As you probably know, the set of all Gibbs states associated to a given interaction potential is a simplex. In particular, all its elements can be written in a unique way as a convex combination of the extremal Gibbs measures. The extremal measures can be argued to be the physically relevant ones, and since the other measures do not contain interesting additional physics, the important point is that all extremal Gibbs measures can be obtained as limits of finite-volume Gibbs measures. In fact, if $\mu$ is an extremal Gibbs measure, then you can use as a boundary condition $\mu$-almost any configuration.

*There are models in which all Gibbs measures can be obtained as limit of finite-volume ones, but these should be the exceptions (when there are more than one Gibbs measure, of course). An example is the two-dimensional Ising model (and the same is probably true for general two-dimensional Potts models, but there are a few things that should be checked). 

*I am not sure that (necessarily non-extremal) Gibbs measures that are not limits of finite-volume ones are rare. For example, consider the four-dimensional Ising model. Then, I have difficulties imagining boundary conditions that will yield most nontrivial mixtures of $\mu^+$ and $\mu^-$. In lower dimensions, you can use the fact that (all in two dimensions, some in three dimensions) interfaces have unbounded fluctuations to create such mixtures by choosing suitable Dobrushin-type boundary condition. But interfaces are expected to have bounded fluctuations in dimensions $4$ and higher.

*In all the above, I am assuming the usual case of  determinsitic boundary conditions. If you allow random boundary conditions sampled from arbitrary probability measures, then you can obviously write any Gibbs measure as the limit of finite-volume ones. This more general setting is not used very often nowadays, but see, for example, Section 2.1.6 in Presutti's book.
