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

## TL;DR

This paper proves that under certain scaling and neutrality conditions, Gibbsian ensembles of 2D Euler point vortices converge to Gaussian distributions, revealing Gaussian fluctuations around the mean field limit.

## Contribution

It establishes a Central Limit Theorem for Gibbsian invariant measures of 2D Euler equations, connecting vortex ensembles to Gaussian fluctuations.

## Key findings

- Convergence of vortex ensembles to Gaussian distributions.
- Validation of the Central Limit Theorem in this context.
- Analysis of partition functions and Gaussian measures.

## Abstract

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. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.