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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 Nāā. 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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# Bose-Einstein condensation in the canonical ensemble

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