Rudolf Clausius was both a mathematician and physicist. Sadi Carnot was a mechanical engineer. Lord Kelvin did not spare a crown, and became a specialist of all the three disciplines. These people are usually regarded as the “inventors” of thermodynamics. Nevertheless, it seems to me that chemists have a better grasp on thermodynamics than most physicists. When I got out of the university as a brand new theoretical nuclear physicist, I knew very little thermodynamics. Indeed, temperature and pressure have not much to say in nuclear physics, unless you are interested in the inner working of star cores. And I must say, thermodynamics had always been some kind of mystery to me, in the early years. I struggled making sense of the first law written in terms of work and heat: pressure, volume and temperature were so crystal clear concepts to me, as much as “work” and “heat” smelled of archaic, almost alchemic contraptions, so vague and undefined. It was only much later that Giorgio Mazzone, a chemist turned physicist who was my group leader at the Materials Chemistry Division in ENEA, Rome, taught me the essentials of thermodynamics. We as a group worked on the physical chemistry of solid-state metallic alloys, and it was there that eventually physics and chemistry could meet, and a common language could be established. In our never-ending and animated discussions, I eventually learned much more about entropy by Giorgio, a rather mercurial scientist who could be exceedingly friendly the morning, and treat you like the sole of his shoes the afternoon. One stimulating way of learning thermodynamics was through its many paradoxes, from Maxwell’s demon, to Loschmidt’s paradox, to more applied issues like the Kauzmann’s paradox that puzzled our discussions about glass-forming metals. And obviously, the Gibbs paradox: a curiosity that has kept many bright minds thinking and fighting for decades.
The classical Gibbs paradox takes the form of a thought experiment involving a box with a partition that separates two bodies of gas. When the partition is removed, the two gases mix spontaneously. To an informed observer who can distinguish the two gas types, the system’s entropy appears to increase. On the other hand, for an ignorant observer who cannot discern any differences between the two gas particles, there is no visible mixing, and the entropy remains unchanged. However, this is not merely a “difference of opinion”, but has a physical significance, since if the entropy increases one could extract work through the mixing process. But the system’s entropy should be an objective quantity – something that does not depend on the different outcomes for the two observers. Gibbs had already noted that the extraction of work depends on the experimental apparatus of the observer, therefore it is something relative to our experience. If the observer knows something about the system, he can build a machine to exploit such a knowledge; if, on the other hand, he knows nothing, he cannot build any machine and no work is extracted.
In more formal terms, the entropy of mixing is the sum of the logarithms of the concentrations. If we have NA particles in the first gas and NB particles in the other gas, with N=NA+NB and xA=NA/N, xB=NB/N, it is 𝛥S= –k(NA ln xA + NB ln xB) > 0. If we now imagine any difference (mass, charge, chemistry, interaction…) between particles A and B to vanish, the mixing entropy must change discontinuously from a finite value to zero. If the entropy were a property of the system (reality) instead of a description of the system (representation of our knowledge), a discontinuity of the entropy would be very strange (what about its derivatives?). However, if the entropy is given by the Boltzmann probability, which is, in turn, related to our knowledge of the system, then there would be no problem with our description (i.e., knowledge) of a system changing discontinuously, when our information content also changes discontinuously. If we cannot determine experimentally that there are two different types of particles, then a description that lumps them together will still be correct. For example, it is common practice to lump the various isotopes of an element together for most thermodynamic applications: although different isotopes are clearly distinguishable, the macroscopic predictions are not affected. The problem of discontinuity is often expressed in terms of a change from distinguishable to indistinguishable particles (hence the famous 1/N! normalizing factor that apparently Gibbs added to his formula for the entropy, on a purely heuristic basis). However, such a change is intrinsically discontinuous, and could not occur simply because the interactions between the particles become identical. So, the problem remains: is entropy a reality, or a trick of our perception?
Recently, a theoretical paper in Nature Communications (https://www.nature.com/articles/s41467-021-21620-7 ) added a quantum layer of knowledge to the Gibbs’ paradox. In their work, Benjamin Yadin, Benjamin Morris and Gerardo Adesso, all at the University of Nottingham, considered a toy model of an ideal gas with non-interacting quantum particles, distinguishing the two gases by a spin-like degree of freedom. They described the mixing processes as it could be performed by both informed and ignorant observers, taking into account their different levels of control, and could calculate the corresponding entropy changes and thus the work extractable by each observer. For the informed observer, the same results as obtained by classical statistical mechanics arguments are recovered. However, for the ignorant observer, there is a marked divergence from the classical case. Counter-intuitively, the ignorant observer can typically extract more work from distinguishable gases – even though they appear indistinguishable – than from truly identical gases. In the continuum and large particle number limit, which classically recovers the ideal gas, this divergence is maximal: the ignorant observer can extract as much work from apparently indistinguishable gases as the informed observer. The analysis hinges on the symmetry properties of quantum states under permutations of particles. For the ignorant observer, these properties lead to non-trivial restrictions on the possible work-extraction processes. Viewed another way, the microstates of the system described by the ignorant observer are highly non-classical entangled states, in particular their number count is quite larger than in the classical case. This implies a fundamentally different way of counting microstates, and therefore computing entropies, compared to what is done classically. Therefore, these findings represents a genuinely quantum thermodynamical effect in the Gibbs mixing scenario, a result impossible in classical physics, as it relies on the symmetry requirements of bosons and fermions. One interesting feature of the results, is that the work extraction protocol for the ignorant quantum observer is necessarily not deterministic: for each value of total spin J, a different amount of work is extracted with probability pJ. This fluctuation behavior is typically expected in thermodynamics of small systems; however, in classical macroscopic thermodynamics, such fluctuations should become negligible. The theoretical results show that the mean value of extracted work grows logarithmically, whereas fluctuations tend to a constant, for both bosons and fermion systems. Therefore, work extraction tends to be fully deterministic even for the case of apparently indistinguishable quantum gases, since fluctuations die out. Experiments are proposed for example on Bose-Einstein condensates, from which an effective quantum heat engine could be devised to operate in regimes where a classical heat engine would fail.
To conclude on a lighter note (after all, this is my “fun physics” newsletter), I will share with you a little-known story about Gibbs’ family that, if has little to do with physics, singularly links to a great movie that I presume most of you readers watched a few years ago: Amistad, about the revolt of slaves on a Spanish ship in the mid XIX century. Gibbs’ life had a quiet, uniform aspect, void of any anecdotes, or those diverting (or perverting) traits that are often associated with genius. He spent his childhood, undergraduate, and graduate years entirely in New Haven, Connecticut, just a few miles from his birthplace. After receiving his degree, he traveled with his sisters for about three years in Europe, during which he could attend many lectures from the greatest scientists of the time. Then, he quietly returned to the family home, where he lived for the rest of his life teaching mathematical physics at Yale, never married, and a careful financial manager (at his death he left an estate worth about 3 million of today’s dollars). However, the year he was born, 1839, was marked by an unusual family event. His father, Josiah Willard sr., professor of theology at the same Yale university, was an active abolitionist. He got interested in the case of the 53 African slaves who had mutinied on the Amistad ship in July. He visited the prisoners, who could not communicate since they spoke only an unknown African language. Gibbs’ father had the idea of learning himself how to count to ten in their language, by using pennies as units. Then, he went on the moors of New Haven harbor, speaking out loud the numbers from 1 to 10 in the African language (which turned out to be the mende of Sierra Leone), until he found a couple of sailors who understood the words. He recruited the two as interpreters and took them back to the court in Hartford, where they could help the prisoners to defend their case, and get finally freedom. Gibbs’ father later published also vocabularies of the mende and other African languages.