The mathematics of the ensemble theory
Abstract
This study shows that the generalized Boltzmann distribution is the only distribution mathematically consistent with thermodynamics when the system is described by an ensemble of a certain mathematical form. This mathematical form is very general, such that the canonical, grandcanonical, or isothermalisobaric ensemble theories are all special cases of this form. Compared with the standard textbook formalism of the statistical mechanics (SM), this approach does not require a prior distribution, does not assume the functional form or maximization of entropy, and employs fewer assumptions. Therefore, this new insight challenges the belief on the requirement of a prior distribution in SM and provides a new way to derive the Boltzmann distribution. This study also reveals the logical and mathematical constraints of SM's fundamental components; therefore, it could potentially benefit researchers on nonBoltzmannGibbs SM and philosophers studying the foundations of SM.
 Publication:

Results in Physics
 Pub Date:
 March 2022
 DOI:
 10.1016/j.rinp.2022.105230
 Bibcode:
 2022ResPh..3405230G
 Keywords:

 Statistical mechanics;
 Boltzmann distribution;
 Ensemble theory