Abstract
We consider the mean field equation based on Onsager's equilibrium statistical mechanical theory for a point vortex system in a two-dimensional bounded domain, in order to understand the two-dimensional turbulence with large-scale long-lived vorticity structures. We show the rigorous derivation of the mean field equation for the system where each point vortex has a unique circulation value and the point vortex number density on the circulation is determined by a probability measure. We make use of the fact that a limiting canonical measure in the mean field limit is regarded as a solution of the variational problem related to the minimum free energy principle.
