The Micro-Canonical Point Vortex Ensemble: Beyond Equivalence

The fluid limit N → ∞ is constructed for a sequence of ensembles of N classical point vortices in a finite domain A ⊂ ℝ² whose ensemble densities (w.r.t. Liouville measure) are Gaussian approximations to δ(E - H). Letting the variance → 0 after N → ∞ has been taken, one recovers the special class of nonlinear stationary Euler flows that is expected from the microcanonical ensemble. The construction improves over previous ones which either had to regularize the logarithmic singularities of the point vortex Hamiltonian or had to assume equivalence of ensembles. In particular, nonequivalence between micro-canonical and canonical ensemble prevails for certain geometries where conditionally stable configurations with negative 'global vortex pair-specific heat' can and do exist in the micro-canonical but not in the canonical ensemble.