When it comes to classical versus quantum statistical mechanics the main difference is that quantum particles are fundamentally indistinguishable. You can't label them and you can't distinguish them by following their trajectory, because the notion of a trajectory becomes meaningless. All this becomes relevant for many particles especially when they interact. The total wave function of the many particles system is not just a product of single particle wave functions, but an expression adopted to what is called the exchange symmetry. That is the property which tells you how the phase of a wavefunction changes when you exchange coordinates. Nature realized two particular phase changes which give rise to Fermions (minus sign on particle exchange) and Bosons (plus sign). At high temperature this kind of entenglement which can even be there just due to exchange symmetry without interaction, becomes less and less important and your system will indeed obey classical statistics. At low temperature (what high and low means is determined by the thermal wave length relative to the average distance between the atoms) the effects of this kind of entenglement are important and drastic differences to classical behavior appear. For examples, electrons behave like a degenerate gas (a Fermi gas which only resembles classical behavior for temperature much higher than the Fermi temperature determined by the Fermi energy aka chemical potential. For regular metals the Fermi temperature is far above evaporation of the material, so metals are allways quantum). Bosons can condensate into a new state of matter known as the Bose-Einstein condensate. Textbooks are not always clear about the fact that summation over independent particles is already a classical (high temperature) approximation even if one still has, e.g., spin states in mind. For example the textbook solution of paramagnetism is a case where we of course sum over independent particles. Strictly speaking we neglect an important aspect of the quantum features. That is rarely explicitly expressed. At extremely low temperatures this may give rise to corrections. One can estimate those temperatures. Because spins for example are attached to atoms, those temperatures are extremely low.

When it comes to classical versus quantum statistical mechanics the main difference is that quantum particles are fundamentally indistinguishable. You can't label them and you can't distinguish them by following their trajectory, because the notion of a trajectory becomes meaningless. All this becomes relevant for many particles especially when they interact. The total wave function of the many particles system is not just a product of single particle wave functions, but an expression adopted to what is called the exchange symmetry. That is the property which tells you how the phase of a wavefunction changes when you exchange coordinates. Nature realized two particular phase changes which give rise to Fermions (minus sign on particle exchange) and Bosons (plus sign). At high temperature this kind of entenglement which can even be there just due to exchange symmetry without interaction, becomes less and less important and your system will indeed obey classical statistics. At low temperature (what high and low means is determined by the thermal wave length relative to the average distance between the atoms) the effects of this kind of entenglement are important and drastic differences to classical behavior appear. For examples, electrons behave like a degenerate gas (a Fermi gas which only resembles classical behavior for temperature much higher than the Fermi temperature determined by the Fermi energy aka chemical potential. For regular metals the Fermi temperature is far above evaporation of the material, so metals are allways quantum). Bosons can condensate into a new state of matter known as the Bose-Einstein condensate. Textbooks are not always clear about the fact that summation over independent particles is already a classical (high temperature) approximation even if one still has, e.g., spin states in mind. For example the textbook solution of paramagnetism is a case where we of course sum over independent particles. Strictly speaking we neglect an important aspect of the quantum features. That is rarely explicitly expressed. At extremely low temperatures this may give rise to corrections. One can estimate those temperatures. Because spins for example are attached to atoms, those temperatures are extremely low.

Dear Christian,

I think your answer covers most of the points asked by Manish Khare, I have no issues with that, but I'd like your opinion on the following topic. You state: "At high temperature this kind of entanglement which can even be there just due to exchange symmetry without interaction, becomes less and less important and your system will indeed obey classical statistics." I have the following problem. In statistical mechanics we have the so-called Gibbs paradox, which in virtually all physical chemistry textbooks is resolved by stating that a factor 1/N! in the partition function, needed to give entropy the correct properties, is in fact a remnant of the quantum mechanical symmetry. I don't believe this explanation is the correct one, but that is not the point I want to make or discuss. My question is: if your statement is correct, how does this factor 1/N! survive the high temperature limit (or doesn't it?), and is there any way I can see how the "notion of a trajectory" for individual distinguishable particles becomes meaningful? I think I've collected most of the literature around the Gibbs paradox, but I don't remember seeing a discussion of this point. If what you state is correct, and in the high temperature limit we indeed get distinguishable trajectories, shouldn't the symmetry then also disappear? Which would show in yet another way that the textbook resolution of the paradox is in fact incorrect.

Dear Gert, those questions are not easy and most of the time answers are not as straight forward to see as we would like. I would need a more careful look to my records to address it in more detail, but let me get started with a few remarks. My thoughts here are not nicely organized so I will jump around a bit.

Indeed the N! will survive the high T-limit. This may not be so surprising, because it is an oversimplification to say that in the high T-limit the world behaves classically. In this context, I would not feel comfortable with stating that one should assign meaningful trajectories to atoms. In fact, I would argue one should always try to derive the apparent classical behavior as a limit of the quantum description where trajectories are not meaningful. Some people would say it is not so much surprising that the world is fundamentally quantum, it is rather surprising that there are these classical observations which ultimately all must originate from quantum mechanics. Having that said one may argue that in the high T-limit particles have high kinetic energy and thus large momentum and therefore deBroglie wavelengths which are small in comparison with the distance between atoms. Therefore overlap of the wavefunctions will be small and one may look at atoms as particles more or less with some trajectory. Similarly, it is not hard to show for interaction free quantum particles that for both a Boson gas and a Fermi gas in the limit of high temperatures the equation of state becomes the good old PV=nRT. The problem why I can't give a simple way to show that the N! in the partition function of the canonical ensemble truly originates from the indistinguishably is that I, like most texts, do not use the canonical ensemble to derive the thermodynamics of interaction free quantum particles. I rather use the grandcanonical ensemble, because it is much easier. In the appendix of our text soon to appear with Wiley we will actually discuss also the approach via the canonical ensemble where the N! is easier to see. Because the N! appears as 1/N! in front of the partition function it is intuitive that it will not go away through a high temperature series expansion. Let me come back to the Gibbs paradox. As you know, the entropy has historically been introduced by Clausius without reference to any microscopic interpretation. The equation dQ=TdS allows to calculate the entropy change from the reversibly exchanged heat for specific equilibrium state changes. The problem here is that only entropy changes can be calculated. For resolving the Gibbs paradox one needs an entropy expression where the entropy of the reference state is explicitly given. I think it is due to this absolute form of the entropy in the Sackur-Tetrode expression for the entropy that the "memory of the quantum origin of the expression" is never erased even at high temperatures. For most practical problems, however, where only entropy changes need to be considered the entropy expression without a term making reference to the particle number works just fine. It may be also interesting to note that it is the same term which contains Planck's constant and one cannot simply perform the limit h->0. There is no way around that a state in phase space needs to occupy a finite volume. There is no meaningful limit to let this volume go to zero. Therefore, strictly speaking, there is no way to do truly classical statistical mechanics. Folks like Gibbs had to introduce already a finite volume for a state in phase space (simply for dimensional reasons there needs to be a constant in front of the phase space integral which makes the partition function just a number). They just did not know at the time that the h in h^3N is the Planck constant.

thank you Christian,

My query is resolved.

and Thanks to Gert too, for highlighting the issue even further.

Thank you.

Dear Christian, thanks for your long and insightful reply. If I understand you correctly you state that in the high temperature limit (I avoided the term "classical limit" since I also don't think they are the same, or even close) the 1/N! survives because you don't really go to (meaningful) trajectories.

Regarding some of your other statements (such as the use of the grand canonical ensemble): I think most of them were made rather forcibly (as was his style) by van Kampen in the attached paper. He makes some interesting comments regarding the Gibbs paradox in the final paragraph. In a 1921 paper of Ehrenfest and Trkal (his ref. 6) this point is also made explicitly: "A truly insightful determination of the N dependence is only possible if we change the value of N by a reversible process and subsequently compare the change in probability with the resulting entropy difference" (emphasized in the paper; my translation). By the way, Ehrenfest and Trkal also mention that Sackur and Tetrode actually had different values for the N dependence: N^{-N} (Sackur) and 1/N! (Tetrode). Of course these are the same for very large N by the Stirling approximation. I have not yet been able to get the original papers by Sackur and Tetrode to see how they arrived at those results, although I can imagine it: both functions of N give a "correct" N dependence in the entropy to make it extensive.

The problem is not just that you can only calculate entropy differences, as you state, but that in order to measure it you need a reversible path between the two states along which you can measure the heat entering or leaving the system. My "sort of" solution to the Gibbs paradox is that you cannot reversibly insert a partition if you want to keep track of the individual trajectories, you can only do that if you don't care which of the atoms are at either side, in which case you also get your 1/N!, without ever resorting to quantum mechanics. This would be in line with van Kampen's final sentence: "In principle therefore, the Gibbs paradox is no different in quantum mechanics, it is only less manifest".

On the other topic you mention: I always viewed the h^3 as the smallest volume of phase space, needed to get finite results for the number of possible realizations. If there is no finite volume element, this number is always infinite, and in the end the size of the finite volume element drops out anyway for measurable quantities. As with the N! (or other possible functions of N) I am not sure if quantum mechanics really resolved that issue and I am also not aware of literature that discusses this point in detail. But I have not really looked for that either. The way you wrote it down suggests that had they thought more deeply at the time about this factor, quantum mechanics should have been anticipated, something along the lines of: "we need a finite volume element to get finite answers for an intermediate step, so in reality there must be a finite volume element." I also don't see how taking the high temperature limit in the sum over states helps. It indeed turns the partition function sum into an integral, but not into an integral over phase space (unless maybe you go through some Wigner distribution first). Also, the N! is irrelevant for the derivation of pV = NkT, isn't it? But, in all honesty, I have to think a little more about your final paragraph (and look up some of my own notes), in particular your implication that there is in fact no truly classical world, we'll always have signatures of the deeper quantum world, before making rash statements like the above.

Would it be possible for me to have a sneak preview of your appendix?

In any case, Manish, I don't think your query was at all stupid, it touches on some of the deepest, and in my opinion mostly unresolved, issues of statistical physics. Even a seemingly minor issue like the Gibbs paradox has been generating papers (and sections in books) for more than 100 years now (last one I know of was three months ago), and I expect it to continue for a while, even though arguments are often repeated, and many papers claim the paradox has "now" been resolved, or was never there, or was something else entirely.

During my afternoon bicycle ride I suddenly remembered an anecdote about stupid questions and van Kampen, told to me a long time ago. I forgot by whom and I don't know if it is true of course, but I think it could be. I also don't know who the speaker in the story is. There are at least a few people on RG who must have known van Kampen personally, maybe they can enlighten me. I only knew him (he died about six months ago at 92) from lectures and classes, but have no bad experiences with him personally.

Nico van Kampen, who became rather famous for his work on stochastic processes in chemistry and physics, could be rather caustic towards students, or even to other experts in his field, when he thought they made foolish remarks, disagreed with him, or asked questions he thought stupid. One time he picked up a person who was supposed to give a talk at a colloquium in Utrecht. On the ride from the train station to the university building they were talking about the lecture, and van Kampen asked the speaker if he had any special requests. Well, there was one thing, the lecturer said, after I've given my talk at question time I'd like a specific question to be asked, since that will give me the opportunity to explain another detail of the work I am presenting. So, if no one else asks that question, could you please ask it? And indeed after the talk question time comes, some questions are asked, and at the end, when there are no more questions from the public van Kampen says something like: "Ok, if no one else has any further questions, I still have one", and proceeds to ask the question he was told to ask. Upon which the lecturer, finally taking revenge for some earlier occasion, says: "that is the most stupid question I ever heard."

Gert,

Surely no one has ever been _that_ cantankerous!

Dear Christian,

It took some digging in my brain and database to find a quote I was thinking of when I wrote my remark about the volume element in phase space, but here it is:

"Therefore it might appear that the shortcoming of classical statistical mechanics was not any failure to yield an extensive entropy function, but only its failure to determine the numerical value of the chemical constant. But even this criticism is not justified at present; the mere fact that it is believed to involve Planck's constant is not conclusive, since e and c are classical quantities and Planck's constant is only a numerical multiple of e^2/c. We see no reason why the particular number 137.036 should be forbidden to appear in a classical calculation. Since the problem has not been looked at in this way before, and nobody has tried to determine that constant from classical physics, we see no ground for confident claims that it can or cannot be done." [p. 16]

Later in the paper, which I attached, Jaynes claims that quantum mechanics does not really solve the problem of the chemical constant either, since there still is an undetermined constant that can multiply h. According to google scholar the paper was cited 103 times so far, but I did not see any titles or abstracts that suggests this particular suggestion was picked up or discussed further.

Gert, I will come back to your discussion at some point. I am currently under terrible time constraints. Midst of June that should be better and I can answer more educated,

Gert, I said I would come back to the discussion. Looking back to what we discussed, I am no longer sure that the thought I had today is what you were actually interested in, but I write it down anyways.

A question I found interesting from what I think emerged from our discussion reads "is there an easy way without referring to urn models to show that the correct Boltzmann counting factor 1/N! in the classical partition function of the canonical ensemble originates from quantum indistinguishability".

What I can suggest it the follow approach, although I must admit that I have not done yet the actual calculation. I suggest to start as usually from the partition function of the grandcanonical ensemble for, e.g., independent fermions. With the help of this partition function one should calculate the internal energy and the entropy. With these results one should be able to calculate F=U-TS which in turn allows to calculate the partition function of the canonical ensemble via Z=e^(-F/kT). In the high temperature limit (T>>E_F/k) this partition function should converge towards the partition function of classical ideal gas and should include the correct Boltzmann counting prefactor. Due to construction I do not expect to see the actual N!,  but rather an approximation of N! which arises from the Stirling formula for ln N! and becomes exact in the thermodynamic limit where ensemble equivalence holds. My understanding is that this must work although I don't know how hard it is to actually do it. I may give it a try if I find time (which is code for not going to happen).

Also, one certainly can directly calculate the entropy expression for an ideal quantum gas. The easiest way to do that is via the grandcanonical partition function. Again, in the limit of very high temperature this entropy expression must converge to the entropy of the ideal gas and it better takes care of the Gibbs paradox. Again, I have not done that, but I have shown that the equation of state of ideal quantum gases (Bosons and Fermions) converges in the limit of very high temperatures (ridiculously high for electrons in metals so that it is rarely considered as a useful limit) towards the good old PV=nRT. 

Quantum mechanics is about the physics of very small things, molecules and smaller. Classical mechanics is about macroscopic things. Quantum mechanics covers the whole of classical mechanics as well, but in the macroscopic limit both become equivalent.