Plea for the use of the exact Stirling formula in statistical mechanics

TL;DR

This paper advocates for using Stirling's exact formula in statistical mechanics to improve the accuracy and consistency of solutions to classical paradoxes like Gibbs' paradox and the extensivity paradox.

Contribution

It demonstrates that employing Stirling's exact formula reintroduces fluctuations, enhancing the rigor of existing solutions to well-known statistical mechanics puzzles.

Findings

Using Stirling's exact formula accounts for fluctuations.

Improves consistency of solutions to Gibbs' paradox.

Resolves the extensivity paradox more rigorously.

Abstract

In statistical mechanics, the generally called Stirling approximation is actually an approximation of Stirling's formula. In this article, it is shown that the term that is dropped is in fact the one that takes fluctuations into account. The use of the Stirling's exact formula forces us to reintroduce them into the already proposed solutions of well-know puzzles such as the extensivity paradox or the Gibbs' paradox of joining two volumes of identical gas. This amendment clearly results in a gain in consistency and rigor of these solutions.