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.