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Bose-Einstein condensates originate with Bose cueing Einstein on the importance of statistics. There are 2 + 1 papers from Einstein on this topic: These papers have been published in proceedings of the meetings of the Prussian Academy of Science, the Sitzungsberichte der Preußischen Akademie der Wissenschaften. One can find them in a book compiling these proceedings edited by Dieter Simon. Scans are also provided by the University of Munster. English translations are available for the first paper and the second paper.
The first two papers are those usually recognized as providing the theory of BEC, in particular predicting the accumulation of particles in the ground state. The third paper was trying to justify the results without recourse to what was actually the most important: the statistics. It appears that neither Einstein nor Bose were aware of the link to indistinguishability, with Einstein, in particular, looking at the "statistical dependence" of Bose as an assumption. For Ehrenfest and others, such results were outright qualified as "disgusting".
Pérez and Sauer[1] give the following chronology of the important developments:
Einstein's theory was for the ideal (non-interacting) gas. The idea soon emerged, however, that interactions are crucial for the phenomenon to take place. As it is my belief that this is not the case, I am interested in tracing the claims and justifications for this preconception.
Gardiner and Zoller[2] want to add to the "simplicity of their quantum-optical treatment":
a realistic treatment of the interatomic interactions which are known to play a major role in the dynamics of the condensing system [41].
Ref. [41] is Griffin's book. On page 2914, they describe quantum solutions for the non-interacting case (in terms of wavelet creation operators).
Jaksch's followup [3] demands interactions for equilibration:
equation that describes N particles interacting with each other by two-particle collisions. These collisions are responsible for the equilibration process.
Nozières in Griffin's book:
do the condensate particles accumulate in a single state? Why don't they share between several states that are degenerate, or at least so close in energy that it makes no difference in the thermodynamic limit? Some Comments on Bose-Einstein Condensation 17The answer is non-trivial: it is the exchange interaction energy that makes condensate fragmentation costly. Genuine Bose-Einstein condensation is not an ideal gas effect: it implies interacting particles!
If interactions are so important, is it not perplexing that the weakly-interacting gas is so close to an ideal BEC? In the case of superconductivity, where interactions are crucial, one does not have a scheme where the weaker the interactions, the closer to a non-interacting system: instead, something qualitatively different happens (with arbitrarily small interactions).