The "Real" Gibbs Paradox and a Composition-Based Resolution

There is no documented evidence to suggest that J. W. Gibbs did not recognize the indistinguishable nature of states involving the permutation of identical particles or that he did not know how to justify on a priori grounds that the mixing entropy of two identical substances must be zero. However, there is documented evidence to suggest that Gibbs was puzzled by one of his theoretical findings, namely that the entropy change per particle would amount to kBln2 when equal amounts of any two different substances are mixed, no matter how similar these substances may be, and would drop straight to zero as soon as they become exactly identical. The present paper is concerned with this latter version of the Gibbs paradox and, to this end, develops a theory characterising real finite-size mixtures as realisations sampled from a probability distribution over a measurable attribute of the constituents of the substances. In this view, two substances are identical, relative to this measurable attribute, if they have the same underlying probability distribution. This implies that two identical mixtures do not need to have identical finite-size realisations of their compositions. By averaging over composition realisations, it is found that (1) fixed composition mixtures behave as homogeneous single-component substances and (2) in the limit of a large system size, the entropy of mixing per particle shows a continuous variation from kBln2 to 0, as two different substances are made more similar, thereby resolving the "real" Gibbs paradox.