On Gibbs properties of transforms of lattice and mean-field systems

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The Gibbsian description of an infinite statistical mechanical system involves prescribing local laws, for a system with a huge number of interacting components, that are consistent with the global behaviour of the system. The system is said to be in phase transition if there is more than one global behaviour compatible with the local laws governing the system. The local laws are usually modeled by conditional distributions in finite regions of the system conditionally on the rest. In most applications there is an energy function (or an interaction) associated with the system and the conditional distributions modeling the local laws are obtained via the Boltzmann-Gibbs weights associated with the energy function. The Gibbs property of a statistical mechanical system is essentially a continuity property of the conditional distributions modeling the local laws cease to be continuous. This discontinuity is caused by the occurrence of a phase transition in some intermediate (internal) system.
The present thesis attempts to answer the question "what are the qualitative properties of initial Gibbsian systems and local transformation (transformations applied independently to the components of the initial system) that will give rise to transformed systems that are Gibbs?" This question is addressed for two classes of statistical mechanical systems. The first class, called lattice systems, consists of systems with components interacting with one another essentially in a local fashion. The second class called mean-field systems covers systems with interaction among all the components.

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