  The second Tisza-Callen postulate of equilibrium thermodynamics states that
for any system exists a function of the system's extensive parameters, called
entropy, defined for all equilibrium states and having the property that the
values assumed by the extensive parameters in the absence of a constraint are
those that maximize the entropy over the manifold of constrained equilibrium
states. By analyzing the evolution of systems of positive and negative absolute
temperatures, we show that this postulate is satisfied by the Boltzmann formula
for the entropy and is violated by the Gibbs formula. Therefore the Gibbs
formula is not a generally valid expression for the entropy.
  Viceversa, if we assume, by reductio ad absurdum, that for some thermodynamic
systems the equilibrium state is determined by the Gibbs' prescription and not
by Boltzmann's, this implies that such systems have macroscopic fluctuations
and therefore do not reach thermodynamic equilibrium.
