The Stumbling Block of the Gibbs Entropy: the Reality of the Negative Absolute Temperatures

The second Tisza-Callen postulate of equilibrium thermodynamics states that for any system there exists a function of the system 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. Based on the thermodynamic evolution of systems which (in the Boltzmann description) have positive and negative temperatures, we show that this postulate is satisfied by the Boltzmann formula for the entropy and may be violated by the Gibbs formula, therefore invalidating the later. Vice versa, 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 the thermodynamic equilibrium.