Next, let us find the occupancy (mean occupation number) fk,ms:
}}{z_{\mathbf{k},m_s}})
.
Now, we note that for |x|<1,
^2})
.
This tells us that the sum in the numerator is thus }}{\left(1-e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}\right)^2})
, and so
}}{z_{\mathbf{k},m_s}}\&=&\frac{\frac{e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}}{\left(1-e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}\right)^2}}{\frac{1}{1-e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}}}\&=&\frac{e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}}{1-e^{-\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}}\f_{\mathbf{k},m_s}&=&\frac{1}{e^{\beta\left(\epsilon_{\mathbf{k}}-\mu\right)}-1}\end{eqnarray})
.
Compare this to the occupancy for a fermion fluid }+1})
; the difference is a change of sign in the denominator. Note that unlike the fermion case, for a boson fluid, the occupancy can be greater than one.
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