Gibbs Paradox - The Mixing Paradox

The Mixing Paradox

A closely related paradox is the mixing paradox. Again take a box with a partition in it, with gas A on one side, gas B on the other side, and both gases are at the same temperature and pressure. If gas A and B are different gases, there is an entropy that arises due to the mixing. If the gases are the same, no additional entropy is calculated. The additional entropy from mixing does not depend on the character of the gases. The paradox is that the two gases can be arbitrarily similar, but the entropy from mixing does not disappear unless they are the same gas.

The resolution is provided by a careful understanding of entropy. In particular, as explained concisely by Jaynes, there is an arbitrariness in the definition of entropy.

A central example in Jaynes' paper relies on the fact that, if one develops a theory based on the idea that the two different types of gas are indistinguishable, and one never carries out any measurement which detects this fact, then the theory will have no internal inconsistencies. In other words, if there are two gases A and B and we have not yet discovered that they are different, then assuming they are the same will cause no theoretical problems. If ever an experiment is performed with these gases that yields incorrect results, we will certainly have discovered a method of detecting their difference and recalculating the entropy increase when the partition is removed.

This insight suggests that the idea of thermodynamic state and entropy are somewhat subjective. The differential increase in entropy (dS), as a result of mixing dissimilar element sets (the gases), multiplied by the temperature (T) is equal to the minimum amount of work we must do to restore the gases to their original separated state. Suppose that the two different gases are separated by a partition, but that we cannot detect the difference between them. We remove the partition. How much work does it take to restore the original thermodynamic state? None – simply reinsert the partition. The fact that the different gases have mixed does not yield a detectable change in the state of the gas, if by state we mean a unique set of values for all parameters that we have available to us to distinguish states. The minute we become able to distinguish the difference, at that moment the amount of work necessary to recover the original macroscopic configuration becomes non-zero, and the amount of work does not depend on the magnitude of that difference.

This line of reasoning is particularly informative when considering the concepts of indistinguishable particles and correct Boltzmann counting. Boltzmann's original expression for the number of states available to a gas assumed that a state could be expressed in terms of a number of energy "sublevels" each of which contain a particular number of particles. While the particles in a given sublevel were considered indistinguishable from each other, particles in different sublevels were considered distinguishable from particles in any other sublevel. This amounts to saying that the exchange of two particles in two different sublevels will result in a detectably different "exchange macrostate" of the gas. For example, if we consider a simple gas with N particles, at sufficiently low density that it is practically certain that each sublevel contains either one particle or none (i.e. a Maxwell-Boltzmann gas), this means that a simple container of gas will be in one of N! detectably different "exchange macrostates", one for each possible particle exchange. Just as the mixing paradox begins with two detectably different containers, and the extra entropy that results upon mixing is proportional to the average amount of work needed to restore that initial state after mixing, so the extra entropy in Boltzmann's original derivation is proportional to the average amount of work required to restore the simple gas from some "exchange macrostate" to its original "exchange macrostate". If we assume that there is in fact no experimentally detectable difference in these "exchange macrostates" available, then using the entropy which results from assuming the particles are indistinguishable will yield a consistent theory. This is "correct Boltzmann counting". It is often said that the resolution to the Gibbs paradox derives from the fact that, according to the quantum theory, like particles are indistinguishable in principle. By Jaynes' reasoning, if the particles are experimentally indistinguishable for whatever reason, Gibbs paradox is resolved, and quantum mechanics only provides an assurance that in the quantum realm, this indistinguishability will be true as a matter of principle, rather than being due to an insufficiently refined experimental capability.