Bose-Einstein condensation in the canonical ensemble

In summary: Note, that he has a factor \sqrt{2} too much in his definition of \tilde{a}_0, but that's not essential]. Then he goes to the thermodynamic limit and finally let the symmetry breaking term go to zero. He clearly states, that this is a purely mathematical trick and that the physical case of an interacting Bose gas is not treated. He then compares the result with the exact solution for the interacting Bose gas. It is not possible to eliminate the fluctuations without the symmetry breaking term. It is possible to calculate the fluctuations in the limit of the symmetry breaking term going to zero. So I think, that this is the right method to treat the case of a non-inter
  • #1
JK423
Gold Member
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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. [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 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?

Thank you in advance for any help you can give me.
John
 
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  • #2
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
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
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
JK423 said:
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?

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
DrDu said:
Bogoliubovs method is not restricted to interacting gasses.
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..
 
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  • #7
Page 138 ff
 
  • #8
Thank you DrDu.
Previously you proposed,
DrDu said:
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.
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
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.
 

1. What is Bose-Einstein Condensation in the Canonical Ensemble?

Bose-Einstein condensation in the canonical ensemble refers to the phenomenon where a large number of bosons (particles with integer spin) become confined to the lowest energy state at low temperatures. This occurs when the thermal energy is less than the energy gap between the ground state and the first excited state.

2. How is Bose-Einstein Condensation different in the canonical ensemble compared to the grand canonical ensemble?

In the canonical ensemble, the number of particles is fixed, whereas in the grand canonical ensemble, the number of particles is allowed to vary. This allows for the possibility of having a non-zero chemical potential, which can affect the behavior of the system and the occurrence of Bose-Einstein condensation.

3. What are the conditions for Bose-Einstein Condensation to occur in the canonical ensemble?

Bose-Einstein condensation in the canonical ensemble occurs when the temperature is low enough for the thermal energy to be less than the energy gap between the ground state and the first excited state. Additionally, the number of particles in the system must be large enough for the Bose-Einstein condensate to form.

4. How does Bose-Einstein Condensation in the canonical ensemble relate to superfluidity?

Bose-Einstein condensation in the canonical ensemble is closely related to superfluidity, which is the property of a fluid to flow without any resistance. In fact, the superfluid state is often considered to be a manifestation of Bose-Einstein condensation in a macroscopic system.

5. Can Bose-Einstein Condensation in the canonical ensemble be observed in real systems?

Yes, Bose-Einstein condensation in the canonical ensemble has been observed in various systems, including ultracold gases of bosonic atoms and exciton-polariton condensates in semiconductor systems. However, the conditions required for Bose-Einstein condensation to occur are very specific, making it a rare phenomenon in most systems.

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