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Let us suppose the gas is confined by a harmonic potential. The bosons have, in three dimensions, energy levels $\hbar\omega(n+3/2)$ with degeneracy $n(n+1)/2$. The grand-canonical partition function of level $n$ is (without degeneracy) $$\xi_n=\sum_{p=0}^\infty \left(\mathrm e^{-\beta \hbar\omega(n+3/2)+\beta\mu(T)}\right)^p$$ where $p$ is the number ...

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"Classical particle", or atoms do condensate. At high temperature $T$, the atoms are far away in the so called gas phase. When the temperature decrease, they will undergo a phase transition and condense to liquid. At even lower temperature, it becomes solid and the atoms are closer together. All of these three phase has clear phase transition temperature. ...

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they can, it's called a regular old liquid. a BEC is when individual atoms lose their identity, in a sense, and the wave functions overlap to display group behaviour. bad for BEC if the particles condense classically and form a liquid = pool of something other than a BEC, prolly just a liquid pool of classical particles. at least as I understand it.

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