Grand canonical ensemble

Remarks

Equivalently, $\Xi = \sum_{N} z^N Z_N(\beta)$, where $z = e^{\beta \mu}$ is the activity and $Z_N$ is the canonical partition function with $N$ particles. For classical particles in a box, $Z_N$ is the canonical integral with the factor $1 / (N! \, h^{3N})$. Example: a single site that is empty ($N = 0$, energy $0$) or holds one particle ($N = 1$, energy $\varepsilon$) has $\Xi = 1 + e^{-\beta (\varepsilon - \mu)}$. Its mean occupation is $\langle N \rangle = 1 / (e^{\beta (\varepsilon - \mu)} + 1)$. Non-example: for one bosonic mode of energy $\varepsilon$ with $\mu > \varepsilon$, the series $\sum_{N \ge 0} e^{-\beta (\varepsilon - \mu) N}$ diverges, so no grand canonical ensemble exists. The grand potential is $\Phi = -\beta^{-1} \ln \Xi$, and for a homogeneous system $\Phi = -pV$, where $p$ is the pressure.