  Despite well over a century of effort, the proper expression for the
classical entropy in statistical mechanics remains a subject of debate. The
Boltzmann entropy (calculated from a surface in phase space) has been
criticized as not being an adiabatic invariant. It has been suggested that the
Gibbs entropy (volume in phase space) is correct, which would forbid the
concept of negative temperatures. An apparently innocuous assumption turns out
to be responsible for much of the controversy, namely, that the energy $E$ and
the number of particles $N$ are given exactly. The true distributions are known
to be extremely narrow (of order $1/\sqrt{N}$), so that it is surprising that
this is a problem. The canonical and grand canonical ensembles provide
alternative expressions for the entropy that satisfy all requirements. The
consequences are that negative temperatures are thermodynamically valid, the
validity of the Gibbs entropy is limited to increasing densities of states, and
the completely correct expression for the entropy is given by the grand
canonical formulation. The Boltzmann entropy is shown to provide an excellent
approximation in almost all cases.
