  In recent papers, several authors have claimed that a definition of the
thermodynamic entropy in terms of the logarithm of a volume in phase space,
originally suggested by Gibbs, is the only valid definition. Arguing from the
Gibbs entropy, these authors claim that thermodynamics cannot be extended to
include negative temperatures. However, the Gibbs entropy fails to satisfy the
postulates of thermodynamics, leading to serious errors. In particular,
predictions of the Gibbs entropy for systems with non-monotonic energy
densities are incorrect. We show that the correct expression for the
equilibrium entropy contains an integral over a surface in phase space, and
negative temperature is a valid thermodynamic concept.
