The Gibbs paradox and the distinguishability of identical particles

Abstract

Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the micro-level, and from the relation S = k ln W leads to a non-extensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable after all; and even that quantum mechanics is indispensable for making sense of this. In contrast, we argue that the classical statistics of distinguishable particles and the resulting non-extensive entropy function are perfectly acceptable from both a theoretical and an experimental perspective. The inconsistency with thermodynamics can be removed by taking into account that the entropy concept in statistical mechanics is not completely identical to the thermodynamical one. We observe that even identical quantum particles are in some cases distinguishable, and conclude that quantum mechanics is irrelevant to the Gibbs paradox.

Unnormalized probability: A different view of statistical mechanics

“…(24) that I had criticized, they resorted to a new statistical mechanics calculation for the composite system. 13 My objection remains valid.…”

“…12,13 Their approach results in a separate calculation in statistical mechanics for every combination of interacting systems. This becomes clear from their response to my statement that the traditional expression for the entropy of a classical ideal gas found without the factor of 1/N!,…”

Ensemble Equivalence for Distinguishable Particles

“…Its physical justification is usually attributed by several authors and textbooks [1,2] to quantum mechanics and the indistinguishable nature of identical particles. The inclusion or not of Boltzmann's correct counting and the correct definition of entropy for distinguishable particles has been widely discussed in the literature since the earlier works by Ehrenfest and Trkal [3] and van Kampen [4] and the more modern contributions, see, e.g., [5][6][7][8][9][10][11][12][13][14].…”

Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy

“…The entropy of mixing and the Gibbs paradox, however, is out of the scope of this work because the problem has recently been solved for distinguishable particles without any ad hoc correction [8,17]. In this respect, the present work tries to generalize the idea to any other chemical transformation of any kind, where symmetry changes might take place.…”

“…The solution of the so-called Gibbs paradox [3][4][5][6][7][8], is probably the most famous example where the same kind of correction has been applied. Gibbs proposed an ad hoc reduction in the entropy of an N -particle system by the amount −k B ln N !, where k B is Boltzmann's constant.…”