The Gibbs Paradox and Particle Individuality

Abstract

Abstract:A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from "yes" to "no" is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S = k ln W or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.

On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures

“…Whether one thinks of ubiquitous substances such as air or drinking water or more specific fluids such as petroleum or milk, they all count as multicomponent systems, i.e., they all comprise more than one identifiable type of constituent. Furthermore, following early commentaries (see [ 1 ] and references therein) on Gibbs’ original work [ 2 , 3 ] on mixtures, the latter are thought to play a key role in the—quantum [ 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 ] or classical [ 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ]—foundations of classical statistical mechanics; through the (in)famous Gibbs paradox . Somewhat surprisingly then, the vast majority of statistical mechanics textbooks covers principally single component systems.…”

Mixing indistinguishable systems leads to a quantum Gibbs paradox

“…These states are degenerate in energy, such that the Hamiltonian of the particles vanishes. This might seem like an unrealistic assumption; however, this model contains the purely combinatorial (or ‘state-counting’) statistical effects, first analysed by Boltzmann 22 , that are known to recover the entropy changes for a classical ideal gas 8 , 23 , 24 using the principle of equal a priori probabilities. One could instead think of this setting as approximating a non-zero Hamiltonian in the high-temperature limit, such that each cell is equally likely to be occupied in a thermal state.…”

Transition rates of non-interacting quantum particles from the Widom insertion formula

“…From (20), it can be seen that transition rates depend only on the number of particles in the destination level, n 2 . If the condition that W n1 ,n 2 depends only on n2 is assumed, then statistics of fermions, bosons, and classical particles should be obtained.…”

“…is obtained in the dilute limit of quantum statistical mechanics. There are other possible justifications that include the assumption that identical classical particles should be treated as permutable; more extensive discussions on this fundamental subject can be found in, for example, [19][20][21][22]. In any case, the statistical weight factor is 1 over the number of distinguishable configurations, n!.…”