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So I understand the Gibbs paradox and that without the $N!$ factor entropy isn't extensive.

But why are these seemingly unrelated things related to each other, why did gibbs in a sense stumbled upon the physical concept of indistinguishability of particles in this way.

If entropy is a measure of microstates; why would distinguishable particles remove extensivity of the microstates? Is there a clear way of understanding this, without big calculations involving Sackur Tetrode equations and such?

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The first question is: Why should entropy be extensive? There might be interesting philosophical discussions about it but, from a practical point of view, if entropy isn't extensive, statistical physics is very boring.

If you look at the partition function and consider that the entropy grows faster than the volume, then, effectively, a large system is always at infinite temperature.

$$ Z\left[\beta\right] =\sum_{\left\{ \sigma\right\} }e^{-\beta E\left[\sigma\right]} =\int d\varepsilon e^{-N\left(\beta\varepsilon-s\left(\varepsilon N\right)\right)} $$

So, at this point, you have two options. Either you completely disregard statistical physics and the concept of temperature (which is hard because it matches experiments) or you conclude that entropy needs to be extensive. Notice that it also can't grow slower than the volume otherwise, the system would always be at a zero temperature.

So, your only option is to model reality in a way that is compatible with this.

Coming back to your question. When you are modelling a system of particles you can make the choice of treating them either as distinguishable or indistinguishable. And it could have been the case that both options were possible. But reality tells you only one of them is right.

(Edit)

Indistinguishability and indistinguishability

There are two different notions of indistinguishability at play, that we need to distinguish. There is the colloquial notion of indistinguishability and the physical notion of indistinguishability. The former tells you that you should consider two electrons distinguishable if they have different "labels" (imagine that each electron has a unique serial number written on its "surface"). The second tells you that you should consider them distinguishable if swapping an electron with a serial number with another electron with a different serial number changes the physics. The statistical physics paradigm is, almost by construction, blind to the microscopic details.

Let's take an example. Consider that there are two different species of electrons, which have different masses (electron1 and electron2). Then, you distinguish them because if you have an electron1 colliding with a wall, the force it will exert on the wall will be different from that of electron2 moving at the same speed. However, if they have the same mass, how can you know the difference?

The point is not if the particles are distinguishable in an absolute sense, the point is if the physics changes when I change one particle by another particle. If, despite having the same mass, you have some other way to distinguish them (e.g. electric charge) then they are still only distinguishable if that difference in charge is relevant to the model at hand.

In other words, what is an electron? Electrons are all the possible particles (in the sense that they may be distinguishable from the perspective of more fundamental and still unknown physics or that could be distinguishable because they carry some unique serial number that is irrelevant for physics), that have a set of properties that are relevant for physics. And because they have the same physical properties, all electrons are indistinguishable.

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    $\begingroup$ Your last paragraph is, at best, misleading. You treat (classical, in principle distinguishable particles like colloids) as indistinguishable because you cannot and do not want to treat them as distinguishable. So whether or not particles should be treated as distinguishable in the context of classical statistical mechanics is a matter of experimental control and knowledge. $\endgroup$ Commented Oct 18, 2023 at 16:18
  • $\begingroup$ Thank you for your answer @JGBM but you have not answered my question. I understand that entropy is extensive; but the core of my question is: why does this suggest indistinguishability of the particles? $\endgroup$ Commented Oct 19, 2023 at 13:46
  • $\begingroup$ @TobiasFünke, I think that we are saying the same thing. What I was trying to say is that there is no a priori physical reason to treat them as indistinguishable, except for the fact that you are forced to do it by reality. There is nothing that prevents me from modelling the particles as being distinguishable, I just won't be able to match any experiments. $\endgroup$
    – JGBM
    Commented Oct 19, 2023 at 23:55
  • $\begingroup$ @bananenheld, because if you want the entropy to be extensive, there is no other option. The factor of N! you have if you treat particles as distinguishable implies that the entropy grows as N log(N) and as such it wouldn't be extensive. $\endgroup$
    – JGBM
    Commented Oct 19, 2023 at 23:59
  • $\begingroup$ No, we are not saying the same thing. As I said, e.g. colloids are distinguishable and yet you treat them as indistinguishable (in macroscopic experiments). The only reason to do so is due to our inability to distinguish them for practical reasons. "Reality" does not force you to do so. I recommend Jaynes' article about the Gibbs Paradox, to start with. $\endgroup$ Commented Oct 20, 2023 at 1:00
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Basically, Gibbs simply had the genius idea, that indistiguishability provides a feasible solution.

I mean, macroscopic thermodynamics was known in practice and consisted of a bunch of empirically established "laws". Many people have been working on providing a microscopic basis for these macroscopic laws.

Please, be aware, that Gibbs solution is not the only possible solution. It is a feasible solution for Maxwell-Boltzmann statistics, but Bose-Einstein and Fermi-Dirac are other solutions (where indistiguishability is already somewhat more inherent), which coincide only (and in some sense) in the classical limit.

In fact, Max Planck found the key towards quantum mechanics by dealing with this bunch of problems for black body radiation. In this sense, we could say somehow, that Max Planck found a "different" solution than Gibbs. :-)

In general, extensivity is only obtained in the thermodynamic limit. Not necessarily in the microscopic model!

Moreover, your question is in some sense logically the wrong direction. You cannot ask for a macroscopic foundation of microscopic Statistical Mechanics. It's rather the other way round: Microscopic Statistical Mechanics is postulated, and macroscopic thermodynamics is then derived via the thermodynamic limit. Microscopic Statistical Mechanics (or Quantum Statistical Mechanics) is a postulate. You can provide intuition for it, motivations for it, but basically, it is simply postulated.

Of course, you can discuss macrosopic consequences of changing some microscopic properties. But most changes of postulated conventional microscopic properties would then contradict (macroscopic) experimental results, and that's the true reason and criterion in (natural) science.

Maybe, Gibbs was just drunk or was on drugs, when he suddenly had that idea. He postulated it, and it worked.

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