Alternative Derivation of the Partition Function for Generalized Ensembles

TL;DR

This paper presents a pedagogical derivation of the partition function emphasizing the role of entropy, clarifying connections between different entropy formulations, and naturally deriving various statistical ensembles from thermodynamic principles.

Contribution

It introduces a new pedagogical approach that derives the partition function and ensembles from entropy considerations, clarifying their fundamental connections.

Findings

Unified derivation of partition functions from entropy principles

Clarification of the relationship between different entropy formulas

Natural emergence of ensemble conditions from thermodynamics

Abstract

A pedagogical approach for deriving the statistical mechanical partition function, in a manner that emphasizes the key role of entropy in connecting the microscopic states to thermodynamics, is introduced. The connections between the combinatoric formula $S= k \ln W$ applied to the Gibbs construction, the Gibbs entropy, $S = -k \sum\limits_i p_i \ln p_i$, and the microcanonical entropy expression $S= k \ln \Omega$ are clarified. The condition for microcanonical equilibrium, and the associated role of the entropy in the thermodynamic potential is shown to arise naturally from the postulate of equal {\itshape a priori} states. The derivation of the canonical partition function follows simply by invoking the Gibbs ensemble construction at constant temperature and using the first and second law of thermodynamics (\emph{via} the fundamental equation $dE = TdS - PdV + \mu dN$) that incorporate the conditions of conservation of energy and composition without the needs for explicit constraints; other ensemble follow easily. The central role of the entropy in establishing equilibrium for a given ensemble emerges naturally from the current approach. Connections to generalized ensemble theory also arise and are presented in this context.