Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

TL;DR

This paper establishes large deviation principles for different statistical ensembles in models of turbulence, analyzing when their equilibrium macrostates are equivalent or not, thus advancing understanding of statistical mechanics in fluid dynamics.

Contribution

It provides the first comprehensive large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles in turbulence models.

Findings

Large deviation principles are proved for canonical and microcanonical ensembles.

Complete equivalence and nonequivalence results are established at the macrostate level.

The results apply to models including 2D fluid motion, quasi-geostrophic flows, and dispersive waves.

Abstract

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the 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 macrostates for the two ensembles.