How wonderful that we have met with a paradox. Now we have some hope of making progress.

The Gibbs Paradox is one of the topics I discussed in my Finland Lectures as an example of problems that never went away. The paradox was, around 1875, discovered by Gibbs himself, who also proposed a solution. The latest paper I know of claiming a resolution was published in 2014[1]

The idea is simple: two parts of an otherwise isolated container, both with volume $$V$$ and filled with an ideal gas at temperature $$T$$, are separated by a partition. Let’s assume that on the left of the partition we have 1 mole of gas A and on the right 1 mole of another gas B. Both gases have of course a certain entropy $$S_{\rm A}$$ and $$S_{\rm B}$$. When we remove the purple partition, the gases ‘mix’: they both expand to occupy the total volume $2V$. The change in entropy $$\Delta S$$ is easily calculated: since the available volume for each gas doubles, $$\Delta S = 2R\ln 2$$. This is an example of an irreversible process, there is no heat effect, so $$\Delta S > q/T = 0$$. Remember that we can only measure the change in entropy if we can find a reversible path between the initial and final state.

The problem arises if we next take the gases to be the same. In the above argument no use is made of the fact that the gases are different, which means that the same change of entropy is found if the gases are identical. After all, they still both expand to fill the larger volume. This is a severe problem for thermodynamics: state quantities can only depend on the current state and if the above argument were correct the entropy of a gas would depend on its history, or on the number of partitions we put in the container before opening them all. Gibbs solution was to make the entropy an extensive property, which amounts to putting for instance a factor $$1/N!$$ in the $$N$$-particle partition function. But this did not resolve the problem, since the first question then is, of course, where this factor comes from.

Most, if not all, textbooks claim that the factor $1/N!$ comes somehow from quantum mechanics. Let me give a few quotes:

The resolution of this paradox is, in the last instance, another example of the successes of quantum mechanics in getting rid of some unsolved problems of classical mechanics, even though in this case there were strong arguments in favor of the generic densities.

D. ter Haar, Elements of Statistical Mechanics, p. 140.

It is not possible to understand classically why we must divide $$\Sigma(E)$$ by $$N!$$ to obtain the correct counting of states. The reason is inherently quantum mechanical. Quantum mechanically atoms are inherently indistinguishable.

K. Huang, Statistical Mechanics, p. 153.

The resolution of Gibbs paradox lies in quantum mechanics […] Identical particles must be counted in a different way from distinguishable particles (they have different “statistics”). This difference between identical and distinguishable particles persists even in the classical limit and leads to a resolution of the Gibbs paradox.

L.E. Reichl, A Modern Course in Statistical Physics, p. 68

Even though the statements raise many questions: what is the classical limit and how does symmetry survive it, to mention just two, you might think the textbook answers resolve the question. But that is not the case at all. I compiled a list of papers about this paradox (probably far from complete), of which the last one is from 2014. If you read them you’ll notice that many of them claim to finally resolve the paradox. Some claim there is no paradox at all, some that quantum mechanics has nothing to do with it, others that there are in fact more paradoxes (up to three), or that it is a different paradox altogether. I can recommend in particular the papers by Schrödinger, Jaynes, and van Kampen. Here is an interesting quote from one of Jaynes’ papers:

Therefore it might appear that the shortcoming of classical statistical mechanics was not any failure to yield an extensive entropy function, but only its failure to determine the numerical value of the chemical constant. But even this criticism is not justified at present; the mere fact that it is believed to involve Planck’s constant is not conclusive, since $$e$$ and $$c$$ are classical quantities and Planck’s constant is only a numerical multiple of $$e^2/c$$. We see no reason why the particular number 137.036 should be forbidden to appear in a classical calculation. Since the problem has not been looked at in this way before, and nobody has tried to determine that constant from classical physics, we see no ground for confident claims that it can or cannot be done.

E.T. Jaynes, The Gibbs Paradox, p. 17.

As far as I know this suggestion was never investigated further.

Other things to explore: what does “different” mean? What does “distinguishable mean? Why should entropy be extensive? How can we mix/unmix reversibly? Is the “limit” from different to the same (gases) continuous?

[1] H. Peters, Demonstration and resolution of the Gibbs paradox of the first kind, Eur. J. Phys., 35, (2014), 015023.