[pishpash]

It's not just that, right? You make it sound the same as hitting the diffraction limit in optics: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/ray...,

but at the Raleigh criterion and under, the waveforms can still be quite different depending on the source distance, and furthermore you can definitely tell the difference of those waveforms from that of a single source.

What I've read about Bose-Einstein condensates seems to imply that in the condensate form, the probability waves not only become unresolvable but also synchronized in phase, AND the energy behavior of the aggregate is markedly different since they "all" (or at least according to Bose-Einstein statistics) occupy the same quantum state: https://www.youtube.com/watch?v=shdLjIkRaS8

Is the transition from Maxwell-Boltzmann statistics to Bose-Einstein statistics a sharp transition or not? In other words, are condensates a descriptive marker or a suddenly different state?

[VorticesRcool]

I am not sure what you mean by the probability waves (amplitude?) becomes unresolvable for the condensate, but it is true that the condensate will have a continuous phase with integer windings of 2 pi around vortices etc. The wavefunctions of the atoms overlap below the critical temperature and you get Bose-Einstein condensation for atoms with integer spin.

But those atoms themselves are still made up of electrons, protons and neutrons which have half integer spin, and at even smaller scales of quarks and gluons. If you probe the condensate with high enough frequency without thermalizing it you would be able to resolve those details, but at the macroscopic level of the condensate those details are not resolvable (is that what you were getting at?).

When you cool an atomic cloud below a critical temperature there will be a condensate fraction and non condensate fraction. If you are just looking at the condensate fraction then you can use Bose-Einstein statistics.

At zero temperature with 100% of the atomic cloud as condensate ( in reality we can never get to zero temperature, but we can get pretty damn close), the Gross–Pitaevskii equation ( https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equat... ) is a good model for the dynamics of the condensate. If you want to go above zero temperature and include interaction with the thermal cloud (the non-condensate fraction), then you can use the SPGPE, the stochastic projected Gross–Pitaevskii equation.

[pishpash]

I mean "unresolvable" in the same sense that two nearby point sources through a diffraction-limited lens are unresolvable. If you watch the video, it appears as if there is only one wavefunction for the BEC of many particles (presuming that is what the video is depicting). Is that in any way an accurate depiction of what's going on?

[gji]

Indeed, that's accurate (apart from finite temperature and interaction effects). A BEC is in some ways pretty similar to a laser, where all the photons are in the same state, even quantum mechanically.

But it's important to distinguish between a BEC and a superfluid. Superfluids are the substances with strange collective properties, and these properties come from being cold, bosonic, and interacting. BECs with very low inter-particle interactions do not behave like superfluids, but will exhibit e.g. interference (just like a laser, which is kinda like a non-interacting BEC).

[lippel82]

Regarding the transition from Maxwell-Boltzmann to Bose-Einstein statistics: There is no transition. If you're dealing with Bosons, it's Bose-Einstein statistics at any energy. It's just that at higher energies (or, lower densities in phase-space), Bose-Einstein and Fermi-Dirac statistics become indistinguishable, with Maxwell-Boltzmann statistics being their common "high energy" limit.

[rubidium]

It's a phase change, so yes a "suddenly different state". In the lab the BEC part and the thermal part are pretty easy to tell apart when you let the distribution expand (by releasing it from the confining potential.