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

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.