Bose-Einstein condensation of the ideal gas

History

Bose-Einstein condensates originate with Bose cueing Einstein on the importance of statistics. There are 2 + 1 papers from Einstein on this topic:

1. Quantentheorie des einatomigen idealen Gases. A. Einstein in Königliche Preußische Akademie der Wissenschaften. Sitzungsberichte 261-267 (1924).
2. Quantentheorie des einatomigen idealen Gases. A. Einstein in Sitzungsberichte der Preu\SSischen Akademie der Wissenschaften 1:3 (1925).
3. Zur Quantentheorie des idealen Gases. A. Einstein in Sitzungsberichte der Preu\SSischen Akademie der Wissenschaften 1:18 (1925).

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 give the following chronology of the important developments:

• 4 June 1924 Bose writes to Einstein
• ca. 2 July 1924 Bose's paper (translated by Einstein) received by Zeitschift für Physik
• 10 July 1924 Einstein's first paper on QTMIG presented to the Prussian Academy (PA)
• 20 September 1924 Einstein's first paper on QTMIG published (Einstein 1924)
• December 1924 Einstein's second paper on QTMIG signed and Bose's paper published (Bose 1924)
• 8 January 1925 Einstein’s second paper on QTMIG presented to PA
• 29 January 1925 Einstein’s third paper on QTMIG presented to PA
• 9 February 1925 Einstein’s second paper on QTMIG published (Einstein 1925a)
• 5 March 1925 Einstein’s third paper on QTMIG published (Einstein 1925a)

The role of interactions

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 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  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.

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).