Statistical Mechanics of the N-point Vortex System with Random Intensities on R

The system of N -point vortices on R2 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∞, 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: −∆u(x) = C−1 Z (0,1] re−βru(x)−γr|x| 2 P (dr), ∀x ∈ R,