Gibbs Equilibrium Fluctuations of Point Vortex Dynamics

TL;DR

This paper studies the fluctuations of a large system of point vortices in a bounded domain, showing that as the number of vortices grows, the fluctuations follow a generalized 2D Euler dynamics preserving a Gaussian energy-enstrophy ensemble.

Contribution

It proves that the space-time fluctuation field of point vortices converges to a generalized 2D Euler dynamics in the large N limit, extending understanding of vortex fluctuations.

Findings

Fluctuation field converges to a generalized 2D Euler dynamics.

Preservation of Gaussian Energy-Enstrophy ensemble in the limit.

Validates mean field approximation for large vortex systems.

Abstract

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 $\beta\geq 0$. We prove that the space-time fluctuation field around the (constant) Mean Field limit satisfies when $N\to\infty$ a generalized version of 2-dimensional Euler dynamics preserving the Gaussian Energy-Enstrophy ensemble.