We consider canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of $$\mathbb {R}^2$$ 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 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.

中文翻译：

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二维欧拉方程吉布斯不变测度的中心极限定理

我们在二维环面上或 $$\mathbb {R}^2$$ R 2 的有界域中考虑欧拉点涡旋的规范吉布斯系综。我们证明了在涡强度的中心极限标度下，并且假设系统在有界域情况下（中性条件）具有零全局空间平均值，系综收敛到所谓的能量-熵高斯随机分布。这可以解释为描述围绕 Caglioti 等人的涡旋集合的平均场极限的高斯涨落。(Commun Math Phys 143(3):501–525, 1992) 和 Kiessling and Wang (J Stat Phys 148(5):896–932, 2012)，它概括了 Bodineau 和 Guionnet 波动的结果 (Ann Inst H庞加莱概率统计 35(2):205–237, 1999)。主要论点在于证明涡旋配分函数的收敛性。