Electron. J. Diff. Eqns., Vol. 2005(2005), No. 92, pp. 1-26.

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

Cassio Neri

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]} r\hbox{e}^{-\beta ru(x)-
 \gamma r|x|^2}P(\hbox{d}r), \quad\forall x\in \mathbb{R}^2,
where $\displaystyle C  =  \int_{(0,1]}\int_{\mathbb{R}^2}\hbox{e}^{-\beta ru(y)
 - \gamma r|y|^2}\hbox{d} yP(\hbox{d}r)$.

Submitted March 3, 2005. Published August 24, 2005.
Math Subject Classifications: 76F55, 82B5.
Key Words: Statistical mechanics; N-point vortex system; Onsager theory; mean field equation.

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Cassio Neri
Instituto de Matemática
Universidade Federal do Rio de Janeiro
Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, Brazil
email: cassio@labma.ufrj.br

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