Gibbs Paradox and the Fluctuation Theorems

Abstract

In this chapter, we discuss a close relation between the Gibbs paradox and the fluctuation theorem with absolute irreversibility. First of all, we give a brief review of what Gibbs discussed in his early article and later renowned textbook to show that Gibbs himself considered related but distinct aspects of the Gibbs paradox. Then, following the article by van Kampen, we classify issues of the Gibbs paradox into three categories: consistency within thermodynamics, consistency within statistical mechanics and inter-theoretical consistency between thermodynamics and statistical mechanics. Among them, we here concentrate on the last aspect, i.e., the issue to determine the relation between the thermodynamic and statistical-mechanical entropies. As shown by Jaynes, this relation is fixed by the requirement of extensivity for the thermodynamic entropy in the thermodynamic limit. However, this resolution cannot be applied to a small thermodynamic system because extensivity itself no longer holds. We show that, in a small thermodynamic system, the fluctuation theorem with absolute irreversibility takes the place of extensivity. To be more specific, by comparing the entropy production and absolute irreversibility in identical- and different-gas mixings, we reproduce the renowned N! factor between the thermodynamic and statistical-mechanical entropies. Finally, we give related discussions to clarify significance of our work.