Statistical mechanics of two-dimensional vortices in a bounded container

The equilibrium statistical properties of a system consisting of a large number of two-dimensional vortices of arbitrary sign, evolving in an arbitrary domain closed by a bounded curve, are investigated under the fundamental assumptions of the equiprobability of any state compatible with a given value of the energy. The Salzberg and Prager equation of state is proved to be also valid in this general case. An expansion in inverse powers of the density closes the hierarchy of equations deduced from the fundamental assumption. The resulting nonlinear partial differential equation for the one-point probability density function, is solved numerically in various cases. A numerical method of simulation using a regular grid is tested for a small number of vortices. Some very accurate numerical simulations of vortex motion in a circular domain confirm the theoretical conclusions.