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We study Onsager's theory of large, coherent vortices in turbulent flows in the approximation of the point-vortex model for two-dimensional Euler hydrodynamics. In the limit of a large number of point vortices with the energy perpair of vortices held fixed, we prove that the entropy defined from the microcanonical distribution as a function of the (pair-specific) energy has its maximum at a finite value and thereafter decreases, yielding the negative-temperature states predicted by Onsager. We furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy, and, under regularity assumptions, obeys the Joyce-Montgomery mean-field equation. We also prove that, under appropriate conditions, the vorticity distribution is the same as that for the canonical distribution, a form of equivalence of ensembles. We establish a large-fluctuation …