Resolving the debate about proposed expressions for the classical entropy

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.