thermodynamics - What is the resolution to Gibb's paradox?

This question is essentially a duplicate of Gibbs Paradox - why should the change in entropy be zero?. The question concerns the following situation: I have some gas of identical particles and they are in a box which has been partitioned into two halves by a removable divider. Now the question is "If I remove the divider, why should the change in entropy be 0?" The standard answer is given in the above-linked-to question. The main idea of the answer is that the particles are supposed to be considered indistinguishible. If you treat the particles this way, you find that there is no change in entropy when the partition is removed.

I understand this answer, and I suppose it should work for identical atoms, but you can easily imagine a situation where you have a collection of objects which are distinguishible, say some nanoparticles with differing numbers of constituent atoms.

Furthermore, you should be able to treat a gas of identical particles as classical distinguishable bodies, and still get the right answer from statistical mechanics. I would say this is a good check of understanding of statistical mechanics.

If this isn't enough motivation to come up with an alternate resolution to the paradox, consider this paradox, which requires essentially the same resolution that the gibbs paradox does: Imagine two systems initially in thermal contact and in thermal equilibrium. Now suppose they are separated. Here we might think the entropy will decrease in this process because in the initial configuration the energy of both systems were allowed to change as long as their sum remained constant. After separation, the systems' energies are fixed at some value. Clearly the set of final states is a subset of the set of initial states, so the entropy has decreased. This is the analogy of the gibbs paradox where instead of the systems exchanging particle number, they are exchanging energy. I would expect it to have essentially the same resolution.

So my question is, "Why should the change in entropy be zero, even in the particles are distinguishible?"

Answer

So my question is, "Why should the change in entropy be zero, even if the particles are distinguishable?"

In statistical physics, entropy can be defined in many different ways.

One possibility is to define it as log of the accessible phase space, given the macroscopic constraints (volume). Such entropy is not a homogeneous function of energy, volume and the number of particles. If the wall is removed, this entropy increases. See the paper by Veerstegh and Dieks: http://arxiv.org/abs/1012.4111

Another way is to define it as log of the accessible phase space divided by the number of permutations of the particles. This entropy is a homogeneous function of energy, volume and the number of particles. If the wall is removed, this entropy stays the same.

Both these definitions are valid, and lead to different concepts of entropy. There is no "the correct" entropy. However, for convenience it is often preferable to use the second definition. The entropy of two interacting sub-systems in equilibrium is then sum of their corresponding entropies and removal of the wall does not change the entropy, which are very practical conventions expected already from classical thermodynamics.