Comparing Boltzmann and Gibbs definitions of entropy in small systems

The long-standing contrast between Boltzmann’s and Gibbs’ approach to statistical thermodynamics has been recently rekindled by Dunkel and Hilbert, who criticize the notion of negative absolute temperature (NAT) as a misleading consequence of Boltzmann’s definition of entropy. A different definition, due to Gibbs, has been proposed, which forbids NAT and makes the energy equipartition rigorous in arbitrarily sized systems. The two approaches, however, are shown to converge to the same results in the thermodynamical limit. A vigorous debate followed Dunkel and Hilbert’s work, with arguments against and in favor of Gibbs’ entropy. In an attempt to leave the speculative level and give the discussion some deal of concreteness, we analyze the practical consequences of Gibbs’ definition in two finite-size systems: a non-interacting gas of N atoms with two-level internal spectrum, and an Ising model of N interacting spins. It is shown that, for certain measurable quantities, the difference resulting from Boltzmann’s and Gibbs’ approach vanishes as N-1/2, much less rapidly than the 1/N slope expected. As shown by numerical estimates, this makes the experimental solution of the controversy a feasible task.