What is the Gibbs Paradox
The Gibbs' paradox is a term from statistical mechanics and occurs when calculating the entropy of mixing of single-phase substances. It was named after its discoverer Josiah Willard Gibbs.
If two different substances are mixed, the entropy increases as the achievable phase space volume increases. It shouldn't make a difference whether you're looking at the mixture of two different or two identical substances. Therefore, it follows from the formula for the entropy of mixing derived from Gibbs that the entropy should also increase if one mixes two volumes of the same substance.
This assumption is correct according to the classical idea. Each atom would be given a number and one could imagine mixing atoms with even and odd identification numbers. According to today's models, atoms or molecules that consist of the same elementary particles are indistinguishable because they are quantum-mechanically described by the same wave functions, which is why no increase in entropy can be observed when the same substances are mixed. Because of this, the paradox does not appear in modern physics.
Let us now consider a structure that consists of two vessels that are only separated by a partition that can be opened and closed. Furthermore, let the same substance in both vessels be at the same pressure and temperature. The partition can now be opened, which leads to mixing and increases entropy. If you now close the partition again, the initial state is restored: the same substance is again in both vessels with the same pressure and the same temperature.
But now you face a problem. Either one assumes that the entropy has been reduced by closing the partition, which would violate the second law of thermodynamics and would also only work for the same substances, but not for different substances in the initial state. This idea for a solution is absolutely arbitrary and cannot be justified sensibly. If one assumes, however, that the entropy has actually increased in this cycle, the entropy could be increased with this reversible process, which would make the term entropy nonsensical.
To resolve the paradox one has to insert a correction term that compensates for the overcounting of the phase space volume by interchanging identical particles. From a quantum mechanical point of view, such a correction term arises naturally, so that the paradox is thus solved by quantum mechanics. As a result, when two volumes of the same substance are mixed, the phase space volume does not increase and the entropy therefore also remains unchanged. In the case of different substances, however, the particles of one substance are distinguishable from those of the other substance, whereby the phase space volume and with it the entropy continue to increase, which is in line with the expectation that the mixture of different substances is an irreversible process.
In quantum mechanics, the degeneracy of many-particle states through permutation of the particles is known as exchange degeneracy. Observation shows that it occurs in nature no Exchange degeneracy there; that is the content of the exchange postulate. With his considerations on entropy of mixing, Gibbs had come across a very deep-seated principle, which is one of the most important of modern physics.
The Gibbs Paradox - E. T. Jaynes (1996)
J. Willard Gibbs' Statistical Mechanics appeared in 1902. The American physicist E. T. Jaynes comes in an analysis of an earlier text by Gibbs (Heterogenous Equilibrium (1875–78) came to the conclusion that Gibbs himself had already found satisfactory answers there and that the paradox is therefore actually not one. In particular, it indicates the application and scope of the concept of entropy. Quote  (page 6):
"Nevertheless, we still see attempts to" explain irreversibility "by searching for some entropy function that is supposed to be a property of microstate, making the second law a theorem of dynamics, a consequence of the equations of motion. Such attempts, dating back to Boltzmann's paper of 1866, have never succeeded and never ceased. But they are quite unnecessary; for the second law that Clausius gave us was not a statement about any property of microstates. The difference in dS on mixing of like and unlike gases can seem paradoxical only to one, who supposes erroneously, that entropy is a property of the microstate. "
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