BoseEinstein condensation in the canonical ensembleby JK423 Tags: boseeinstein, canonical, condensation, ensemble 

#1
Dec1212, 12:23 PM

P: 381

Hello guys,
I would really need some help on the following problem. Consider a noninteracting & nonrelativistic bosonic field at finite temperature. We are all aware of the fact that such a statistical system is well described by the grandcanonical 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 grandcanonical ensemble, below the critical temperature, gives huge and unphysical particlenumber 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. [itex]\left\langle {N_0^2} \right\rangle  {\left\langle {{N_0}} \right\rangle ^2} [/itex]) 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 grandcanonical 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? Thank you in advance for any help you can give me. John 



#2
Dec1212, 12:48 PM

HW Helper
P: 1,391

Does the GrossPitaevskii 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
Dec1212, 01:01 PM

Sci Advisor
P: 3,377

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
Dec1312, 06:31 AM

P: 381

BoseEinstein condensation in the canonical ensemble
Thank you both for your feedback!
I think both of the approaches you propose, GP eq. and Bogoliubov method, refer to a gas with 2body interactions and not to a noninteracting 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
Dec1312, 06:57 AM

Sci Advisor
P: 3,377

Bogoliubovs method is not restricted to interacting gasses. 



#6
Dec1312, 07:21 AM

P: 381

This is the book i checked, http://en.bookfi.org/book/773282 . Whenever you have time please show me where this method is presented.. 



#7
Dec1312, 07:33 AM

Sci Advisor
P: 3,377

Page 138 ff




#8
Dec1512, 05:52 PM

P: 381

Thank you DrDu.
Previously you proposed, 



#9
Dec1612, 01:47 AM

Sci Advisor
P: 3,377

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. 


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