Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 92, pp. 1-26.

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

Author: Cassio Neri (Univ. Federal do Rio de Janeiro, Brazil)

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