The best answer to the puzzle of Gibbs about $N!$!: A note on the paper by Sasa, Hiura, Nakagawa, and Yoshida

TL;DR

This paper clarifies the derivation of the factorial $N!$ factor in statistical mechanics, extending the minimum work principle to small systems, and offers a new perspective using a process discussed by Horowitz and Parrondo.

Contribution

It provides a self-contained explanation of how the minimum work principle determines the $N!$ factor, connecting classical statistical mechanics with a process from Horowitz and Parrondo.

Findings

The $N!$ factor is uniquely determined by the minimum work principle.

A new perspective on the $N!$ factor derivation using Horowitz and Parrondo's process.

Clarification of the classical Gibbs puzzle regarding factorials in statistical mechanics.

Abstract

In a recent paper [1], Sasa, Hiura, Nakagawa, and Yoshida showed that a natural extension of the minimum work principle to small systems uniquely determines the factor $N!$ that arrises in relations connecting statistical mechanical functions (such as the partition function) and thermodynamic functions (such as the free energy). We believe that this provides us with the clearest answer to the "puzzle" in classical statistical mechanics that goes back to Gibbs. Here we attempt at explaining the theory of Sasa, Hiura, Nakagawa, and Yoshida [1] by using a process discussed by Horowitz and Parrondo [2] in a different context. Although the content of the present note should be obvious to anybody familiar with both [1] and [2], we believe it is useful to have a commentary that presents the same theory from a slightly different perspective. The present note is written in a self-contained manner. We only assume basic knowledge of classical statistical mechanics and thermodynamics. We nevertheless invite the reader to refer to the original paper [2] for background, references, and related discussions, as well as the original thoughts.