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Attempts to establish microcanonical entropy as an adiabatic invariant dateback to works of Gibbs and Hertz. More recently, a consistency relation basedon adiabatic invariance has been used to argue for the validity of Gibbs(volume) entropy over Boltzmann (surface) entropy. Such consistency relationequates derivatives of thermodynamic entropy to ensemble average of thecorresponding quantity in micro-state space (phase space or Hilbert space). Inthis work we propose to re-examine such a consistency relation when the numberof particles ($N$) is considered as the independent thermodynamic variable. Inother words, we investigate the consistency relation for the chemical potentialwhich is a fundamental thermodynamic quantity. We show both by simpleanalytical calculations as well as model example that neither definitions ofentropy satisfy the consistency condition when one considers such a relationfor the chemical potential. This remains true regardless of the system size.Therefore, our results cast doubt on the validity of the adiabatic invarianceas a required property of thermodynamic entropy. We close by providingcommentary on the derivation of thermostatistics from mechanics which typicallyleads to controversial and inconsistent results.

Microcanonical entropy: consistency and adiabatic invariance