Comment on “Quirks of Stirling’s Approximation”

摘要

In the article “Quirks of Stirling’s Approximation” published in this Journal , Macrae and Allgeier appear to conclude erroneously that the exactness of Boltzmann’s entropy formula for the microcanonical ensemble depends on the ensemble size. This conclusion seems to originate from the adoption of an unnecessary approximation appearing in Physical Chemistry: A Molecular Approach by McQuarrie and Simon. This letter provides a simple derivation of the entropy of the microcanonical ensemble that leads to the exact Boltzmann expression. In addition, this letter aims to stress that the entropy of a system or an ensemble is truly maximized only when no constraint is imposed on the system or ensemble. In particular, when the entropy of a microcanonical ensemble is to be maximized, it is inappropriate to assume the most probable distribution of the systems in the ensemble because this assumption is an unnecessary and unjustified constraint.