Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles

We consider a general class of statistical mechanical models of coherent st ructures in turbulence, which includes models of two-dimensional fluid moti on, quasi-geostrophic flows, and dispersive waves. First, large deviation p rinciples are proved for the canonical ensemble and the microcanonical ense mble. For each ensemble the set of equilibrium macrostates is defined as th e set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macro-states for the two ensembles. Microcanonical equilibri um macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are character ized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the foll owing question in global optimization. What are the relationships between t he set of solutions of the constrained minimization problem that characteri zes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and suf ficient condition for equivalence of ensembles to hold at the level of equi librium macrostates is that it holds at the level of thermodynamic function s, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formu lation. If the microcanonical entropy is not concave at some value of its a rgument, then the ensembles are nonequivalent in the sense that the corresp onding set of microcanonical equilibrium macrostates is disjoint from any s et of canonical equilibrium macrostates. we point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We a lso introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complemen tary set microcanonically. Such ensembles arise naturally in applications w here there are several independent dynamical invariants, including models o f dispersive waves for the nonlinear Schrodinger equation. Complete equival ence and nonequivalence results are presented at the level of equilibrium m acrostates for the pure canonical, the pure microcanonical, and the mixed e nsembles.