Mean field theory and coherent structures for vortex dynamics on the plane

TL;DR

This paper derives the Onsager-Joyce-Montgomery equilibrium theory for point vortices on the plane using modern variational methods, linking classical vortex theory with statistical mechanics and large deviation principles.

Contribution

It provides a new derivation of the OJM theory through the Bogoliubov-Feynman inequality, connecting it to mean field and large deviation theories.

Findings

Links classical vortex equilibrium theory to modern variational methods

Shows Landau's approximation as a large deviation principle

Provides a rigorous foundation for the maximum entropy principle in vortex systems

Abstract

We present a new derivation of the Onsager-Joyce-Montgomery (OJM) equilibrium statistical theory for point vortices on the plane, using the Bogoliubov-Feynman inequality for the free energy, Gibbs entropy function and Landau's approximation. This formulation links the heuristic OJM theory to the modern variational mean field theories. Landau's approximation is the physical counterpart of a large deviation result, which states that the maximum entropy state does not only have maximal probability measure but overwhelmingly large measure relative to other macrostates.