  The long standing contrast between Boltzmann's and Gibbs' approach to
statistical thermodynamics has been recently rekindled by Dunkel and Hilbert
[1], 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 arbitrary sized systems. The two approaches,
however, are shown to converge to the same results in the thermodynamical
limit. A vigorous debate followed ref.[1], with arguments against [2,3] and in
favor [4,5,6,7] 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 the
inverse square rrot of N, 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.
