Thomas Kuhn, the famous philosopher of science, envisioned that scientific revolutions take place when “an increasing number of epicycles” arise, resulting in the untenability of a prevailing theory. Just in case you aren’t familiar, the “epicycles” are a reference to the Ptolemaic world-view with the earth at the center of the universe. To explain the trajectories of the other planets, Ptolemaic theory required that the planets circulate the earth in complicated trajectories called epicycles. These convoluted epicycles were no longer needed once the Copernican revolution took place, and it was realized that our solar system was heliocentric.

This post is specifically about the Gibbs paradox, which provided one of the first examples of an “epicycle” in classical mechanics. If you google Gibbs paradox, you will come up with several different explanations, which are all seemingly related, but don’t quite all tell the same story. So instead of following Gibbs’ original arguments, I’ll just go by the version which is the easiest (in my mind) to follow.

Imagine a large box that is partitioned in two, with volume V on either side, filled with helium gas of the same pressure, temperature, etc. and at equilibrium (i.e. the gases are identical). The total entropy in this scenario is . Now, imagine that the partition is removed. The question Gibbs asked himself was: does the entropy increase?

Now, from our perspective, this might seems like an almost silly question, but Gibbs had asked himself this question in 1875, before the advent of quantum mechanics. This is relevant because in classical mechanics, particles are always distinguishable (i.e. they can be “tagged” by their trajectories). Hence, if one calculates the entropy increase assuming distinguishable particles, one gets the result that the entropy increases by .

This is totally at odds with one’s intuition (if one has any intuition when it comes to entropy!) and the extensive nature of entropy (that entropy scales with the system size). Since the size of the larger container of volume containing identical gases (i.e. same pressure and temperature) does not change when removing the partition, neither should the entropy. And most damningly, if one were to place the partition back where it was before, one would naively think that the entropy would return to , suggesting that the entropy decreased when returning the partition.

The resolution to this paradox is that the particles (helium atoms in this case) are completely indistinguishable. Gibbs had indeed recognized this as the resolution to the problem at the time, but considered it a counting problem.

Little did he know that the seeds giving rise to this seemingly benign problem required the complete overthrow of classical mechanics in favor of quantum mechanics. Only in quantum mechanics do truly identical particles exist. Note that nowhere in the Gibbs paradox does it suggest what the next theory will look like – it only points out a severe shortcoming of classical mechanics. Looked at in this light, it is amusing to think about what sorts of epicycles are hiding within our seemingly unshakable theories of quantum mechanics and general relativity, perhaps even in plain sight.

This is an interesting topic, which is quite controversial. Do you know the book “Equilibrium Statistical Mechanics” by E. Atlee Jackson? There is an Appendix “D. Concerning the Entropy Constant”. He argues that the laws of thermodynamics do not require the entropy to be extensive; only changes dS have to be extensive. He disagrees that the Gibbs paradox leads to indistinguishable particles. See also the Wikipedia article on the Gibbs paradox and the references given there.

It would be great if you could further comment on this (maybe a follow up blog post?). This seems to be a rare example where a common textbook explanation is actually wrong. Thanks for bringing this topic up.

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Thank you for pointing this out. I was not aware of the controversy. I will definitely take a closer look at the references and hopefully post an update soon.

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I have always had some problems with understanding Gibbs paradox because I thought that there is no way a purely classical theory(classical statistical mechanics) would be inconsistent with itself and would show the necessity for quantum mechanics.

Reif, in his book uses wrong arguments based on reversibility to persuade the reader.(not quite the two arguments used here)

Recently, I cam across this paper which claims that Gibbs paradox is vacuous. I haven’t had a chance to sift through it. Maybe you could take a look: http://www.mdpi.com/1099-4300/10/1/15

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Thank you also for pointing this out!

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The common view that the Gibbs paradox exhibits an inconsistency of classical (statistical) mechanics and is an early pointer towards quantum mechanics is wrong. The resolution of the Gibbs paradox for classical (distinguishable) particles is given in http://dx.doi.org/10.1088/0143-0807/35/1/015023 (also freely available on arXiv).

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