Abstract
We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.
Highlights
The paper analyses the mean-field limit and the corresponding fluctuations for the point vortex dynamics, at equilibrium with positive temperature, arising from a class of equations generalising the Euler equations
When m = 2, the model corresponds to the Euler equations, and when m = 1 it corresponds to the inviscid surface quasi-geostrophic (SQG) equation
We prove a law of large numbers and, in terms of θ, that the limit is a stationary solution of the original equation
Summary
The paper analyses the mean-field limit and the corresponding fluctuations for the point vortex dynamics, at equilibrium with positive temperature, arising from a class of equations generalising the Euler equations. T2 the with periodic velocity, and boundary conditions and zero spatial average. When m = 2, the model corresponds to the Euler equations, and when m = 1 it corresponds to the inviscid surface quasi-geostrophic (SQG) equation. One route to understand the behaviour of a turbulent flow is to study invariant measures for the above equations. Onsager [43] proposed to do this via a finite dimensional system, called
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