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

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.