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For more than 100 years, one of the central concepts in statistical mechanicshas been the microcanonical ensemble, which provides a way of calculating thethermodynamic entropy for a specified energy. A controversy has recentlyemerged between two distinct definitions of the entropy based on themicrocanonical ensemble: (1) The Boltzmann entropy, defined by the density ofstates at a specified energy, and (2) The Gibbs entropy, defined by the sum orintegral of the density of states below a specified energy. A criticaldifference between the consequences of these definitions pertains to theconcept of negative temperatures, which by the Gibbs definition, cannot exist.In this paper, we call into question the fundamental assumption that themicrocanonical ensemble should be used to define the entropy. Our argument isbased on a recently proposed canonical definition of the entropy as a functionof energy. We investigate the predictions of the Boltzmann, Gibbs, andcanonical definitions for a variety of classical and quantum models, includingmodels which exhibit a first-order phase transition. Our results support thevalidity of the concept of negative temperature, but not for all models with adecreasing density of states. We find that only the canonical entropyconsistently predicts the correct thermodynamic properties, whilemicrocanonical definitions of entropy, including those of Boltzmann and Gibbs,are correct only for a limited set of simple models.

Comparison of canonical and microcanonical definitions of entropy