A Can indistinguishable particles obey Boltzmann statistics

[Philip Koeck]

A little spin-off from this thread: A state for 1 particle is given by a small volume of size h³ in phase space. If two particles occupied the same volume in 1 particle phase space that would mean, in classical terms, that they are at the same spatial coordinates and moving with the same momentum vector at a given time. In other words they would be inside each other. For classical particles (C60-molecules etc.) I would say that's not possible. That seems to indicate that FD statistics is the obvious choice for describing classical particles. Most textbooks, however, introduce classical systems as having no limit for the number of particles per state. Do you agree with my thinking?

 

[vanhees71]

Hm, that's a contradictio in adjecto, because classical particles make only things in a realm, where the Bose or Fermi nature is irrelevant. Both the Bose and the Fermi statistics have as the low-occupation-number limit the Boltzmann statistics (including the $N!$ factor "repairing" the Gibbs paradox). The low-occupation-number constraint makes the indistinguishability of particles irrelevant since there are on average less than 1 particle in a single-particle phase-space cell of size $h^3$.

 

[Philip Koeck]

Hm, that's a contradictio in adjecto, because classical particles make only things in a realm, where the Bose or Fermi nature is irrelevant. Both the Bose and the Fermi statistics have as the low-occupation-number limit the Boltzmann statistics (including the $N!$ factor "repairing" the Gibbs paradox). The low-occupation-number constraint makes the indistinguishability of particles irrelevant since there are on average less than 1 particle in a single-particle phase-space cell of size $h^3$.

Assuming we could create a system of classical particles like C60 molecules at high occupancy, would it follow FD or BE statistics? Or is this not even a sensible question?

 

[vanhees71]

It depends on how high the occupancy is. As a whole C60 is a boson. So if not too close together they behave as bosons. Also the carbon atoms are bosons (if you have the usual $^6\text{C}$ isotope), but of course on the level of the fundamental constituents you have fermions. I guess, however, that to get this fermionic nature into action you's have to pack the buckyballs so close together that you destroy them ;-).

 

[Philip Koeck]

It depends on how high the occupancy is. As a whole C60 is a boson. So if not too close together they behave as bosons. Also the carbon atoms are bosons (if you have the usual $^6\text{C}$ isotope), but of course on the level of the fundamental constituents you have fermions. I guess, however, that to get this fermionic nature into action you's have to pack the buckyballs so close together that you destroy them ;-).

Are you beginning to see the problem? If C60 truly behaved like a boson you would be able to put any number of particles into the same state (or "point" in phase space). I find that really hard to imagine. I think they'll simply and very classically be in each others way, even considering the effects of uncertainty. To me it seems that quantum statistics simply doesn't apply to systems that are "too classical".

 

[DrClaude]

Are you beginning to see the problem? If C60 truly behaved like a boson you would be able to put any number of particles into the same state (or "point" in phase space). I find that really hard to imagine. I think they'll simply and very classically be in each others way, even considering the effects of uncertainty. To me it seems that quantum statistics simply doesn't apply to systems that are "too classical".

Bose-Einstein condensates of molecules exist. While no one has been able to cool molecule as big as C₆₀ down to temperatures where BEC happens, there is no reason to think it doesn't make sense for many C₆₀ molecules to be in the same quantum state.

By the way, double-slit type experiments have been performed using C₆₀ (and even bigger molecules), and quantum effects are visible.

 

[Philip Koeck]

Bose-Einstein condensates of molecules exist. While no one has been able to cool molecule as big as C₆₀ down to temperatures where BEC happens, there is no reason to think it doesn't make sense for many C₆₀ molecules to be in the same quantum state.

By the way, double-slit type experiments have been performed using C₆₀ (and even bigger molecules), and quantum effects are visible.

Thanks. Experiments are always convincing. Maybe it is time to skip all classical statistics and start directly with quantum statistics as vanHees suggested earlier.

 

[Philip Koeck]

One more thing has turned up. It's been mentioned several times (also on Wikipedia and in textbooks) that the Boltzmann distribution is a high temperature and low occupancy limiting case of the BE and FD distributions. I can show that W approaches the correct Boltzmann counting for low occupancy as discussed in posts 69 to 73 (before calculating a distribution), but I'm having a hard time seeing how high T would help in general. Only if I insert expressions for the chemical potential and density of states that are valid for an ideal gas of indistinguishable particles into the BE or FD distribution, I get something that approaches the Boltzmann distribution for high T. Is the mentioned limiting case general or only valid for the ideal gas? Can anyone point me to some literature?

 

[bhobba]

but I'm having a hard time seeing how high T would help in general.

See the following:
https://ps.uci.edu/~cyu/p115A/LectureNotes/Lecture13/lecture13.pdf

The distribution depends on a parameter β which is small at high temperatures. This gives the familiar Boltzmann-Maxwell distribution..

Thanks
Bill