Microcanonical entropy: consistency and adiabatic invariance

Attempts to establish microcanonical entropy as an adiabatic invariant date back to works of Gibbs and Hertz. More recently, a consistency relation based on adiabatic invariance has been used to argue for the validity of Gibbs (volume) entropy over Boltzmann (surface) entropy. Such consistency relation equates derivatives of thermodynamic entropy to ensemble average of the corresponding quantity in micro-state space (phase space or Hilbert space). In this work we propose to re-examine such a consistency relation when the number of particles ($N$) is considered as the independent thermodynamic variable. In other words, we investigate the consistency relation for the chemical potential which is a fundamental thermodynamic quantity. We show both by simple analytical calculations as well as model example that neither definitions of entropy satisfy the consistency condition when one considers such a relation for the chemical potential. This remains true regardless of the system size. Therefore, our results cast doubt on the validity of the adiabatic invariance as a required property of thermodynamic entropy. We close by providing commentary on the derivation of thermostatistics from mechanics which typically leads to controversial and inconsistent results.