Entropies of the microcanonical ensemble

We revisit the Boltzmann entropy and the Gibbs entropy, two entropy definitions of the microcanonical ensemble, and discuss their respective weakness, incompatible with our traditional understanding of thermodynamic entropy. For the microcanonical ensemble specified at energy E, the Boltzmann entropy is determined by the density of states at E, whereas the Gibbs entropy is given by the total number of states having energies not greater than E. The Boltzmann entropy violates the fundamental relations in thermodynamics, and the degree of the violation is of the order of a finite size correction, usually negligible for large systems. Regardless of system size, the Gibbs entropy complies with the thermodynamic relations but breaks the additive property that the thermodynamic entropy of a large system of weakly coupled subparts should equal the sum of thermodynamic entropies of the subparts. We show that, for a traditional setup where a total system consists of a small subsystem and a bath, the additivity breaking is determined by the difference between two temperatures derived from the Boltzmann entropy and the Gibbs entropy. For systems with bound energy spectra and decreasing density of states, the additivity of the Gibbs entropy does not hold even in the thermodynamic limit.