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We know that the ground state is macroscopically occupied. What about the first excited state?

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I am writing this answer in continuation from my last answer on your previously related post. As described there, the macroscopic occupation of the ground state is enforced by the quantum statistics by bounding the number density of excited states. Note that consequently the first excited state is also bounded, and therefore there is a limit to which it can be occupied irrespective of the number density of the system. So there is no similar phenomenon of macroscopic occupation of the excited states here, rather the macroscopic occupation pertains only to the ground state which is free to accommodate as many bosons as possible; which isn't true of the excited states.

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