It is usually seen in literature the terminology that certain bosons proliferate or condense. What is the difference between them? Is it that proliferation just means a large number of bosons occupy certain state, and condensation further means there is also off-diagonal long range order besides that?
For example, in three dimensions free bosons can condense at low temperatures, signaled by a macroscopic number of bosons in the single particle ground state. There is a phase transition going from the high temperature uncondensed phase to the low temperature condensed phase, and this can be seen in singularities in various thermodynamic quantities.
But in two dimensions, as long as the temperature is low enough, there can also be a macroscopic number of bosons in the single particle ground state. However, we do not say this is a Bose-Einstein condensate, because there is no phase transition. Then in this case should we say bosons proliferate but not condense?