| We consider statistical mechanical models of a general class of physical systems for which large-scale, coherent structures persist amidst small-scale, turbulent fluctuations. The main focus of this dissertation is on whether a statistical mechanical formulation using a canonical ensemble yields equivalent predictions to a statistical mechanical formulation using a microcanonical ensemble. For each ensemble, a large deviation principle is proved, and the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. A complete classification of all equivalence and nonequivalence relationships between the sets of canonical equilibrium macrostates and the sets of micro-canonical equilibrium macrostates are given in terms of concavity properties of the microcanonical entropy. The general theory of this dissertation is then applied to the specific, geophysical example of barotropic, quasi-geostrophic turbulence. |
