  The entropy definition in the microcanonical ensemble is revisited. We
propose a novel definition for the microcanonical entropy that resolve the
debate on the correct definition of the microcanonical entropy. In particular
we show that this entropy definition fixes the problem inherent the exact
extensivity of the caloric equation. Furthermore, this entropy reproduces
results which are in agreement with the ones predicted with standard Boltzmann
entropy when applied to macroscopic systems. On the contrary, the predictions
obtained with the standard Boltzmann entropy and with the entropy we propose,
are different for small system sizes. Thus, we conclude that the Boltzmann
entropy provides a correct description for macroscopic systems whereas
extremely small systems should be better described with the entropy that we
propose here.

Statistical Mechanics