Inconsistency of microcanonical entropy: the case of chemical potential

Arash Tavassoli, Afshin Montakhab

Published 2015-12-09Version 1

Recently, there has been a considerable amount of debate regarding a correct and "consistent" definition of microcanonical entropy. Consistent thermodynamics is based on the assumption of adiabatic invariance of microcanonical entropy. Such an assumption is equivalent to a consistency relation which equates derivatives of the entropy to ensemble average of corresponding quantity in micro-state space (phase space or Hilbert space). It has been argued that the Gibbs (volume) entropy satisfies such a consistency relation under general condition, and is thus preferred over the Boltzmann (surface) entropy, which is only consistent for large macroscopic systems. 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 examples that \emph{both} definitions of entropy (Gibbs as well as Boltzmann) lead to inconsistent results when one considers such a relation for the chemical potential for finite or infinite system sizes. 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 thermodynamics from mechanics which typically leads to controversial and inconsistent results. Considering entropy as an \emph{emergent} property of macroscopic systems may provide a way out of such inconsistencies.