Editing Gas in a harmonic trap (section)

=== Massive Bose–Einstein particles ===

For this case:
: <math>\Phi=e^{\beta \epsilon}/z-1</math>
where <math>z</math> is defined as:
: <math>z=e^{\beta\mu}</math>

Integrating the energy distribution function and solving for <math>N</math> gives:
: <math>N = \frac{f}{(\hbar\omega\beta)^3}~\mathrm{Li}_3(z),</math>
where <math>\mathrm{Li}_{s}(z)</math> is the [[polylogarithm]] function. The polylogarithm term must always be positive and real, which means its value will go from 0 to <math>\zeta(3)</math> as <math>z</math> goes from 0 to 1. As the temperature goes to zero, <math>\beta</math> will become larger and larger, until finally <math>\beta</math> will reach a critical value <math>\beta_\text{c}</math>, where <math>z = 1</math> and
: <math>N = \frac{f}{(\hbar\omega\beta_c)^3}~\zeta(3) .</math>

The temperature at which <math>\beta = \beta_{c}</math> is the critical temperature at which a Bose–Einstein condensate begins to form. The problem is, as mentioned above, the ground state has been ignored in the continuum approximation. It turns out that the above expression expresses the number of bosons in excited states rather well, and so we may write:
: <math>N=\frac{g_0z}{1-z}+\frac{f}{(\hbar\omega\beta)^3}~\mathrm{Li}_3(z)</math>
where the added term is the number of particles in the ground state. (The ground state energy has been ignored.) This equation will hold down to zero temperature. Further results can be found in the article on the ideal [[Bose gas]].