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We derive necessary conditions for finite systems to exhibit two locally stable states with respect to fluctuations in the microcanonical and canonical ensembles. The requirements are, respectively, that there be two inflections in the logarithm of the probability distribution of the energy or the number of particles in the canonical or grand canonical ensembles. The thermodynamics associated with the whole phase space may be related to quantities averaged over regions of phase space and are consistent with van Hove's theorem.
Wales et al. (Mon,) studied this question.