I am interested to understand how positive entropy Bose Einstein condensation for cold atoms (say) behave. The way I think about it is as follows: We have an ideal pure state where every atom is in the same ground state which depends on a geometry of a certain trap and the atoms do not interact. Now, naively, I thought about the actual positive-entropy state looking as one of the two following alternatives:
(1) Every atom get excited with some small probability p.
A different picture is as follows:
(2) If the type of the intended ideal state puts every atom in the same ground state A (where A, say, depends on the geometry of the trap), then the positive-entropy state is obtained by mixing this ideal state with a state which is Bose Einstein state w.r.t. a different ground state B (or many such B;s).
I asked around a little and what I was told is that the T>0 theory is not part of the original theory discovered in the 1920s but rather a more recent theory that was developed in the 1990s. Moreover, the perturbation for the ideal pure state described by the T>0 state manifests pairwise interaction between the atoms, and that there are interesting singularities that occur immediately when T>0 no matter how small.
I will be very thankful for some explanations and references. Most helpful for me will be non-technical explanations.
Update: I am aware of one book on a related topic: The Poincare seminar 2003 on Bose Einstein Condensation - Entropy. (I did not get a hold of it yet but a few of the papers are here.) I will appreciate any information regarding the matter. Genneth recommend the book Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems (Oxford Graduate Texts) by Anthony Leggett.
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