The statistical mechanical resolution of a thermodynamic “paradox”

Woodrow Beckford

Yûto Murashita, Masahito Ueda

The Gibbs paradox is often erroneously believed to have been resolved by quantum mechanics. In fact, it has been resolved by the assumption of extensivity only in non-interacting macroscopic thermodynamic systems as implied by Pauli. Yet, this resolution is far from satisfactory not only because real systems are interacting but also because it cannot apply to small thermodynamic systems. Here, we offer the resolution applicable to all of these systems based on a new concept of absolute irreversibility, where extensivity is shown to emerge automatically with increasing the system size. Quite surprisingly, our resolution also removes the fundamental flaw of fluctuation theorems in modern statistical physics.

Chih‐Yuan Tseng

The “Gibbs Paradox” refers to several related questions concerning entropy in thermodynamics and statistical mechanics: whether it is an extensive quantity or not, how it changes when identical particles are mixed, and the proper way to count states in systems of identical particles. Several authors have recognized that the paradox is resolved once it is realized that there is no such thing as the entropy of a system, that there are many entropies, and that the choice between treating particles as being distinguishable or not depends on the resolution of the experiment. The purpose of this note is essentially pedagogical; we add to their analysis by examining the paradox from the point of view of information theory. Our argument is based on that ‘grouping’ property of entropy that Shannon recognized, by including it among his axioms, as an essential requirement on any measure of information. Not only does it provide the right connection between different entropies but, in addition, it draws our attention to the obvious fact that addressing issues of distinguishability and of counting states requires a clear idea about what precisely do we mean by a state.

C. G. Chakrabarti

Thibaut Josset

Hjalmar Peters

The Gibbs paradox of the first kind (GP1) refers to the false increase in entropy which, in statistical mechanics, is calculated from the process of combining two gas systems S1 and S2 consisting of distinguishable particles. Presented in a somewhat modified form, the GP1 manifests as a contradiction to the second law of thermodynamics. Contrary to popular belief, this contradiction affects not only classical but also quantum statistical mechanics. This paper resolves the GP1 by considering two effects. (i) The uncertainty about which particles are located in S1 and which in S2 contributes to the entropies of S1 and S2. (ii) S1 and S2 are correlated by the fact that if a certain particle is located in one system, it cannot be located in the other. As a consequence, the entropy of the total system consisting of S1 and S2 is not the sum of the entropies of S1 and S2.