Abstract.We consider the evolution of a distribution of N identical point vortices when stochastic perturbations in the Hamiltonian are present. It is shown that different initial configurations of vorticity with identical integral invariants may exist. Using the Runge-Kutta scheme of order 4, it is also demonstrated that different initial configurations with the same invariants may evolve without having any tendency to approach to a unique final, axially symmetric, distribution. In the presence of stochastic perturbations, if the initial distribution of vortices is not axially symmetric, vortices can be trapped in certain domains whose location is correlated with the configuration of the initial vortex distribution.

Dans cette Note, nous étudions l'évolution de la répartition de N tourbillons localisés identiques. Nous montrons, en utilisant l'expérience numérique directe, plus précisément le schéma de Runge-Kutta à l'ordre 4, représenté par le modèle des tourbillons ponctuels, que des répartitions initiales, de vorticité différentes avec les mêmes invariants globaux, peuvent exister. Nous montrons que des configurations initiales avec les mêmes invariants peuvent évoluer vers des états quasi-finaux complétement différents.

We formulate nonlinear integro-differential equation for the averaged collective Hamiltonian of a gas of interacting twodimensional vortices, derive its analytical solution, and discuss the equilibrium, axially-symmetrical, probability distributions that are possible for such a model. We also theoretically prove that the probability distribution for a system of 2D point vortices takes a form similar to the Gibbs distribution, but point out that the physical fundamentals of such a system differ from the standard theory of interacting particles. Furthermore, we find thermodynamical functions for positive and negative "temperature" of the system, and point out that the states with positive "temperature" correspond to stationary bell-shape vortex distributions, while the states with negative "temperature" correspond to distributions localized near container walls.