# Bose-Einstein condensation in the canonical ensemble

1. Dec 12, 2012

### JK423

Hello guys,
I would really need some help on the following problem.

Consider a non-interacting & non-relativistic bosonic field at finite temperature. We are all aware of the fact that such a statistical system is well described by the grand-canonical ensemble in the limit N→∞. However, there is a temperature -the critical temperature- below which a quantum phase transition takes place and a finite fraction of the bosons occupy the ground state. Now here's the problem:

I realized that the grand-canonical ensemble, below the critical temperature, gives huge and unphysical particle-number fluctuations in the ground state due to the fact that the latter is occupied by a large number of bosons. In other words, if we calculate the variance of the ground state occupation number (i.e. $\left\langle {N_0^2} \right\rangle - {\left\langle {{N_0}} \right\rangle ^2}$) we will get a really big number comparable to the number of bosons that occupy the ground state. This is a known pathological behaviour of the grand-canonical ensemble for temperatures below the critical and is in sharp contrast to the (correct) canonical ensemble.

For reasons related to the project i am working on, i want to get rid of this pathological behaviour. How can i do that? A first thought is to directly use the canonical ensemble, but then the calculations get to be impossible even numerically! It seems to me that i can, in no way, approach the problem using the canonical ensemble, neither analytically nor numerically.

Do you have any suggestions? Perhaps you're aware of a way working with the canonical ensemble analytically below the critical temperature using a method that i don't know of?

John

2. Dec 12, 2012

### Mute

Does the Gross-Pitaevskii equation suffer from the large particle fluctuations? You might consider using this equation if not, unless there is a particular reason you need to work with the statistical ensembles.

3. Dec 12, 2012

### DrDu

For a precise analysis of the problem, have a look at W. Thirring, Lehrbuch der Mathematischen Physik, Vol. 4, Quantenmechanik grosser Systeme (There exists also an English version) Chapter 2.5.
The solution he describes is known as Bogoliubov method to add an extra symmetry breaking term, perform the thermodynamic limit and then take the limit where the symmetry breaking term goes to 0 in that order.
In praxis this means to replace the anihilation operator for the ground state a_0 by a constant.
This was first worked out by Beliaev.
I found the following new article which should contain relevant references:
http://arxiv.org/abs/1206.5471

4. Dec 13, 2012

### JK423

Thank you both for your feedback!

I think both of the approaches you propose, GP eq. and Bogoliubov method, refer to a gas with 2-body interactions and not to a non-interacting gas. Is that correct?
DrDu i checked Thirring's book, Ch 2.5 but i didn't find anything relevant.. Can you please tell me how the section is called in case i missed it?

5. Dec 13, 2012

### DrDu

Strange. Momentarily I don't have the book at hand. I think I looked for "Bose ..." in the register.

Bogoliubovs method is not restricted to interacting gasses.

6. Dec 13, 2012

### JK423

Really? I'm really interested..!
This is the book i checked,
http://en.bookfi.org/book/773282 [Broken] .
Whenever you have time please show me where this method is presented..

Last edited by a moderator: May 6, 2017
7. Dec 13, 2012

### DrDu

Page 138 ff

8. Dec 15, 2012

### JK423

Thank you DrDu.
Previously you proposed,
as a solution to get rid of the unphysical fluctuations of the grand-canonical ensemble in the condensate phase. I am not sure that's what Thirring does on that page (although i'm not certain because his notation is all so different and confusing). First of all, just replacing the a_0 by a constant doesn't kill the fluctuations in the condensate phase, i tried that. I checked the literature, like Pitaevskii, Leggett and others, and they all say that the only way to kill the fluctuations is to insert 2-body interactions, and then apply the Bogoliubov method you mentioned. Noone reports a way to do all this without introducing interactions so i think it's not possible. Do you agree with all these?

9. Dec 16, 2012

### DrDu

No, I don't agree. I checked yesterday the argument by Thirring.
I meant not only page 138 but also following ones. He explicitly introduces the symmetry breaking term in (2.5,51;5) to use his notation - I hope it coincides with the German edition.