  Very recently, the validity of the concept of negative temperature has been
challenged by several authors since they consider Boltzmann's entropy (that
allows negative temperatures) inconsistent from a mathematical and statistical
point of view, whereas they consider Gibbs' entropy (that does not admit
negative temperatures) the correct definition for microcanonical entropy.
  In the present paper we prove that for systems with equivalence of the
statistical ensembles Boltzmann entropy is the correct microcanonical entropy.
Analytical results on two systems supporting negative temperatures, confirm the
scenario we propose. In addition, we corroborate our proof by numeric
simulations on an explicit lattice system showing that negative temperature
equilibrium states are accessible and obey standard statistical mechanics
prevision.
