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 R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}. 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 Caglioti et al. (Commun Math Phys 143(3):501–525, 1992) and Kiessling and Wang (J Stat Phys 148(5):896–932, 2012), and it generalises the result on fluctuations of Bodineau and Guionnet (Ann Inst H Poincaré Probab Stat 35(2):205–237, 1999). The main argument consists in proving convergence of partition functions of vortices.