| Summary: | In this dissertation we study the asymptotic behavior of two separate models in statistical mechanics. The first is a turbulence model that describes the long-time behavior of a class of dispersive wave equations of nonlinear Schrödinger-type. Numerical studies have shown that, in a bounded domain, solutions of these equations tend in the long-time limit toward a Gibbsian statistical equilibrium macrostate consisting of a ground state solitary wave on the large scales and Gaussian fluctuations on the small scales. A large deviation principle is proved for a mixed Gibbs ensemble that expresses this concentration phenomenon precisely in the relevant continuum limit. The second model investigated is a lattice-spin model due to Blume, Emery, and Griffiths. It is one of the few, and certainly one of the simplest models known to exhibit both a continuous, second-order phase transition and a discontinuous, first-order phase transition. The structure of the sets of equilibrium macrostates with respect to both the canonical and microcanonical ensembles is analyzed for this model. In doing so, we rigorously prove for the first time a number of results that significantly generalize previous studies. For the canonical ensemble, full proofs of the structure of the set of equilibrium macrostates are provided. Using numerical methods and following an analogous technique used in the canonical case, we also analyze the structure of the set of equilibrium macrostates for the microcanonical ensemble.
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