Gibbs "Paradox" and the Entropy of mixing

[rentier]

(not a paradox nowadays, but it was an issue for years)
https://en.m.wikipedia.org/wiki/Gibbs_paradox

The two gases may be arbitrarily similar, but the entropy from mixing does not disappear unless they are the same gas - a paradoxical discontinuity.

It's not a question about a formula. I don't understand the motivation in physics to claim Gibbs mixing "paradox", the discontinuity point. What bothers the physicist to ask for a continuous transition between distinguishable and indistinguishable particles mixing (entropy).
Is it paradox that a sieve separates larger particles and smaller ones? No matter how small the difference is (in theory)? Is it paradox that 0. (real number) is different from any even the smallest number?
Either I can distinguish the particles or not. It's a binary statement/situation. Of course, it's discontinuous by definition. Two different systems; one with distinguishable particles, the other not.
Neither does it clash with the understanding of entropy as a measure of ignorance. If I can somehow differentiate particles, there is a decrease in entropy. I could (more or less laboriously) separate them and reverse the mixing.

I can't catch a logical link to the paradox.

 

[Andy Resnick]

Resolving Gibbs' paradox is as you say- information and entropy are related. ET Jaynes wrote about how Gibbs' paradox is resolved and his papers are quite readable:

https://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf

 

[vanhees71]

The ironic thing is that classical statistical physics is more complicated than quantum statistical physics because of such issues.

One should be aware of the fact that before the "quantum revolution" there was no idea that particles are indistinguishable at all. The reason is simple: Within classical mechanics you can label each individual particle and exactly follow its trajectory. So each particle is individually labelled, e.g., by assigning its initial coordinates at some time $t=0$ to it.

Consequently when Boltzmann invented his "counting method" for microstates, he treated the particles as distinguishable, which lead to the problem with the nonadditivity of entropy and an apparent change of macroscopic state where in fact there is none (with the usual Gibbs paradox, leading to mixing energy where in fact nothing is mixed). Ingenious as Boltzmann was he put another factor $1/N!$ in his counting, i.e., assuming that there's no way to distinguish in the macrostate the individual particles in the container, but that's in fact a clear contradiction to classical mechanis, where the particles are in principle distinguishable. The puzzle thus was, how to justify this additional factor, and the answer was finally given by the modern quantum theory of many-body systems, i.e., the indistinguishability of particles and their bosonic or fermionic nature. The classical limit of course does not lead to distinguishable particles at all, and that solves the problem in saying that you need to keep some quantum aspects also in classical statistical physics, among them the indistinguishability of particles of the same sort. Of course if you have particles of different sort the mixing entropy is real and measurable and also comes out in quantum statistics as it must be.

 

[rentier]

Thanks a lot to you three: Jaynes (tribute), vanhees71, Andy.
I started reading Jaynes, appreciate it so far, interesting and readable, as you recommended, Andy.

@vanhees71: thanks once more for your detailed explanations. I think it is crucial, your remark that the Old Geniuses had just started to explain such subjects. I try to adjust some things in my head, and follow. Thanks.