What even is Gibbs’ paradox?

James Wills

Gibbs' paradox has proved stubbornly persistent in the philosophy and physics literature: after over a century, there is still no consensus on a solution. More worryingly, however, there seems to be no consensus on what the problem is; a combination of many years and the rise of quantum mechanics (and with it, metaphysical speculations on the identity and indistinguishability of particles), have created a menagerie of problems which bear Gibbs' name. The aim of this paper, restricted to the paradox in thermodynamics, is to provide a detailed survey of what textbook authors and philosophers think the paradox is. I will show that there are three distinct puzzles in thermodynamics which claim to be the Gibbs paradox. I argue that the one which is mostly ignored in the literature is the one that was posed and solved by Gibbs in his 1875 - 1878 paper, and that the same paper contains the solutions to the other two paradoxes which subsequently appeared. The outcome of the paper is a clear understanding of how Gibbs understood the entropy of mixing in thermodynamics. I also make some comments about the paradox in statistical mechanics.