A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of $R^2$. We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called Energy-Enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles of cite{clmp92,KieWan2012}, and it generalises the result on fluctuations of cite{bodineau}. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.