The Mathematics of the Ensemble Theory

TL;DR

This paper demonstrates that the generalized Boltzmann distribution is uniquely consistent with thermodynamics within a broad ensemble framework, challenging traditional assumptions in statistical mechanics and offering a new derivation method.

Contribution

It introduces a general mathematical ensemble framework where the Boltzmann distribution is uniquely consistent, removing the need for prior distributions or entropy maximization.

Findings

Generalized Boltzmann distribution is the only thermodynamically consistent distribution.

The approach does not require a prior distribution or entropy maximization.

It encompasses standard ensembles as special cases.

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, grand-canonical, or isothermal-isobaric 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 non-Boltzmann-Gibbs SM and philosophers studying the foundations of SM.