On the thermodynamic entropy in the microcanonical ensemble of classical systems

We demonstrate that the surface entropy given by the volume of an energy shell in the phase space can be the thermodynamically consistent entropy in a classical microcanonical ensemble if the thickness of the energy shell is not an arbitrary constant but a non-extensive function satisfying a specific differential equation. A particular form of the energy shell thickness as a possible solution to the differential equation converts the surface entropy into the volume entropy given by the phase-space volume bounded by a constant energy surface. However, such a form bears a problem: The temperature derived accordingly becomes extensive when the density of states is a non-monotonic function of energy. Based on the adiabatic invariance of the degeneracy of a quantum system and the Weyl correspondence, we propose an alternative solution: the energy shell thickness given by the energy level spacing in the quantum counterpart of the classical ensemble considered, which is illustrated by a few simple examples.