Gibbs Equilibrium Fluctuations of Point Vortex Dynamics

We consider a system of N point vortices in a bounded domain with null total circulation, whose statistics are given by the Canonical Gibbs Ensemble at inverse temperature $\beta \ge 0$. We prove that the space-time fluctuation field around the (constant) Mean Field limit satisfies when $N\to \mathrm{\infty }$ a generalized version of 2-dimensional Euler dynamics preserving the Gaussian Energy-Enstrophy ensemble.