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