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

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 of [9,21], and it generalises the result on fluctuations of [8]. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.

Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

“…The limit Gaussian distribution for the θ variable turns out to be a statistically stationary solution of the equations. The fluctuations result holds due to a higher order expansion analysis of the partition function, similar to [21], where the same statement for Euler vortices has been recently proved.…”

“…Mean-field limit results of point vortices with random intensities can be found in [29,40,41]. The analysis of fluctuations can be found in [3,4] and in the recent [21].…”

Energy conditional measures and 2D turbulence

“…centered Bernoulli random variables, Benfatto et al [3] proved that the canonical Gibbs measures of the point vortices, with appropriately regularized Green functions, converge to the Gaussian measure µ β,γ (dω) = e −βH−γE dω (β, γ > 0, H and E are the energy and enstrophy functionals), which are invariant for the 2D Euler flow. In the recent work [14], analogous result was proved without smoothing the Green function; see [12] for related result concerning the generalised inviscid surface quasi-geostropic equations.…”

Infinitesimal Invariance of Completely Random Measures for 2D Euler Equations

“…This point of view was first considered by Flandoli [14] in the case where M is a Gaussian measure satisfying Hypothesis 1, that is, when M is a multiple of a white noise; in that work it was also discussed how such a notion of solution is a sensible one, producing approximations of Gaussian solutions with smooth solutions and scaling limits of point vortices systems. Related works [19,20] have analyzed more general Gaussian invariant measures, also in cases where the PDE dynamics include stochastic forcing.…”

“…(1.1) were originally introduced and studied in a series of works by Albeverio and others [8,9]: in fact, the idea of Gaussian and Poissonian invariant measures as special cases of the general, independently scattered one, dates back to [7]. The works [10,19,20,17] concern relations, in the form of scaling limits, between Gaussian and Poissonian random fields preserved by Euler's dynamics.…”