Timeline for Grand canonical partition functions for Bose-Einstein statistics vs. Maxwell-Boltzmann statistics
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13 events
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| Apr 13, 2017 at 12:39 | history | edited | CommunityBot |
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| May 14, 2014 at 4:52 | history | edited | joshphysics | CC BY-SA 3.0 |
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| May 13, 2014 at 19:51 | comment | added | joshphysics | @JoshuaMeyers Ah ok. I think I can explain exactly what you're doing by computing the "partition function of a particular state;" I'll think about it a bit more to make sure I understand correctly, and then I'll write an addendum to my answer as soon as I can. | |
| May 13, 2014 at 19:41 | comment | added | Joshua Meyers | I don't know where the factorial factors come from. | |
| May 13, 2014 at 19:40 | comment | added | Joshua Meyers | I think what I'm doing is computing the partition function of a particular state, and then using it to find the average occupancy of that state. I'm treating the state as a system. I got the Bose-Einstein partition function by adding up the Boltzmann factors $e^{-\beta (E-\mu N)}$ for particles in the state (so that $E=N\epsilon$ because all particles in the state have the same energy). I got the Maxwell-Boltzmann partition function from wikipedia (en.wikipedia.org/wiki/…) and though it works to get the average occupancy, | |
| May 13, 2014 at 19:17 | comment | added | joshphysics | @JoshuaMeyers The distribution functions give the ensemble average occupancy of a particular state with energy $\epsilon$. For computing these, you don't need the energies and their degeneracies (see physics.stackexchange.com/questions/101408/…). However, in order to compute the partition function of the whole system, you do need the energies and their degeneracies. It's unclear to me what you're actually computing with the expressions you wrote down for $\mathcal Z$. How did you generate them? | |
| May 13, 2014 at 19:05 | comment | added | Joshua Meyers | Oh I checked my textbook and it says that the distribution functions are for a "system" consisting of one single-particle state, that is in thermal and diffusive equilibrium with the rest of the system. | |
| May 13, 2014 at 19:02 | comment | added | Joshua Meyers | The formulas I have yield the correct results for $\bar{n}=-k_{B}T\frac{\partial}{\partial\epsilon}\ln\mathcal{Z}$. That is why I think they are correct. | |
| May 13, 2014 at 18:57 | comment | added | Joshua Meyers | Yes, they are non-interacting. Do the single-particle energy levels and their degeneracies need to be specified? The derivations of Bose-Einstein and Maxwell-Boltzmann statistics that I have seen don't make explicit reference to them. | |
| May 13, 2014 at 18:27 | comment | added | joshphysics | @JoshuaMeyers Is the system you're considering a system of non-interacting particles? If so, what are the single-particle energy levels and their degeneracies? | |
| May 13, 2014 at 18:02 | comment | added | Joshua Meyers | Or is my expression for Maxwell-Boltzmann wrong? (I am starting to doubt it) | |
| May 13, 2014 at 18:01 | comment | added | Joshua Meyers | So then why is the grand canonical partition function larger when the bosons ARE identical (Bose-Einstein) than when they are not (Maxwell-Boltzmann)? Wouldn't it be smaller, since you are summing over less states? | |
| May 13, 2014 at 17:42 | history | answered | joshphysics | CC BY-SA 3.0 |