  Over the past century, an intense debate in statistical mechanics has been
about the correctness of Boltzmann's surface entropy versus Gibbs' volume
entropy, for isolated systems. Both entropies make significantly different
predictions for systems with few degrees of freedom. Even in the thermodynamic
limit, they can disagree-while Boltzmann entropy allows negative absolute
temperatures to exist, Gibbs entropy precludes such a possibility. Here, we
show that modifying Boltzmann's entropy via a relative energy tolerance
eliminates thermodynamic inconsistencies in several model systems with
unbounded energy spectra by ensuring positive, finite temperatures.
Concomitantly, the proposed entropy allows for negative temperatures in systems
with bounded spectra and closely matches canonical ensemble predictions. This
work conclusively remedies the prevalent deficiencies of the Gibbs and
Boltzmann entropy formulations and paves the way for the use of the modified
Boltzmann entropy in the microcanonical ensemble, allowing negative
temperatures to exist.
