A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container D, as the number of vortices N tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydro- The pdf of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager-Kraichnan condensation in the literature, between low energy states with no mean flow in the domain and high energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of D, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite N , are explored using a Markov chain phase-space sampling method.

The statistics of a neutral point-vortex gas in an arbitrary two-dimensional simply connected and bounded container are investigated in the framework of the microcanonical ensemble, following the cumulant expansion method of Pointin and Lundgren [Phys. Fluids 19, 1459 (1976)]. The equation for vorticity fluctuations, obtained when a thermodynamic scaling limit is taken, is solved explicitly. The solution depends on an infinite sequence of negative "domain inverse temperatures," determined by the domain shape, which are obtained from solutions of a "vorticity mode" eigenvalue problem. An explicit expression for the thermodynamic curve relating inverse temperature and energy is found and is shown to depend on the geometry and not on the scale of the domain. Explicit formulas are then obtained for the time variance of the projection of the vorticity field onto each vorticity mode. The results are verified by two methods. First, for a chosen single-parameter family of domains, direct sampling of the microcanonical ensemble is used to demonstrate the accuracy of the formula for the thermodynamic curve. Second, direct numerical simulations are used to verify the formulas for the variance of the projections of the vorticity field, with convincing results.

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