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Statistical mechanics of the $N$-point vortex system with random intensities on $R^2$

Author(s): Cassio Neri

Journal: Electronic Journal of Differential Equations
ISSN 1072-6691

Volume: 2005;
Issue: 92;
Start page: 1;
Date: 2005;
Original page

Keywords: Statistical mechanics | N-point vortex system | Onsager theory | mean field equation.

The system of N -point vortices on $mathbb{R}^2$ is considered under the hypothesis that vortex intensities are independent and identically distributed random variables with respect to a law $P$ supported on $(0,1]$. It is shown that, in the limit as $N$ approaches $infty$, the 1-vortex distribution is a minimizer of the free energy functional and is associated to (some) solutions of the following non-linear Poisson Equation:$$ -Delta u(x) = C^{-1}int_{(0,1]} rhbox{e}^{-eta ru(x)- gamma r|x|^2}P(hbox{d}r), quadforall xin mathbb{R}^2, $$where $displaystyle C = int_{(0,1]}int_{mathbb{R}^2}hbox{e}^{-eta ru(y) - gamma r|y|^2}hbox{d} yP(hbox{d}r)$
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