Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

Abstract We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations the point vortex model in plane, plasmas and self-gravitating matter 2D, as well higher-dimensional logarithmic interactions appearing conformal geometry power laws. The proofs are based on corresponding “microscopic” concavity result at finite N , shown by leveraging an unexpected link Kähler plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict obtained using uniqueness free minimizers, established companion paper. results imply thermodynamic equivalence ensembles holds this models. As application, it critical inverse negative temperatures—in macroscopic microscopic setting—coincide asymptotic slope microcanonical entropies. Along way, we also extend previous continuous weakly positive definite concerning temperature states, general non-continuous case. In particular, singular situations exhibited where, somewhat surprisingly, fails levels sufficiently close minimum level.