| context | $J\in\mathbb N$ |
| $j\in \{1,\dots,J\}$ | |
| range | $N_j\in \mathbb N$ |
| context | $N_j^\text{max}\in \mathbb N \cup \{\infty\}$ |
| context | $Z_{N_j}(\beta) $ sequences in $N_j$ of canonical partition functions with length $N_j^\text{max}$ |
| definiendum | $\Xi(\beta,\mu_1,\dots,\mu_J):=\sum_{j=1}^J \sum_{{N_j}=0}^{N_j^\text{max}}\ z(\beta,\mu_j)^{N_j}\cdot Z_{N_j}(\beta) $ |
Here $z$ denotes the fugacity. The quantity $J$ denotes the number of different particle species to consider.
The above definition mirrors the classical microcanonical phase volume and the classical canonical partition function.
The summands $z(\beta,\mu_j)^{N_j}\cdot Z_{N_j}(\beta)$ can be viewed as canonical partition function where the distributions $\mathrm e^{-\beta\ H_{N_j}}$ are shifted to $\mathrm e^{-\beta\ (H_{N_j}-\mu_j\ N_j)}$. Accordingly the meaning of the so called chemical potential $\mu_s$ is a package of energy associated with each given particle in the system.
The index $j$ is attached to the particle number $N$ as well as to the chemical potential $\mu$ and is always dropped if the system of interest deals with only a single sort of particle.
Important grand canonical partition functions in QM are those for bosons and fermions denoted $\Xi^+$ and $\Xi^-$, respectively. We only deal with one sort of particle, but introduce the index $r$ which runs over different energy levels. Using the identities
$\sum_{N=0}^{N^\text{max}}({\mathrm e}^{\beta\sum_r \mu})^N {\cdot} {\mathrm e}^{-\beta\sum_r N\varepsilon_r}$
$\sum_{N=0}^{N^\text{max}}(\prod_r{\mathrm e}^{\beta \mu} {\cdot} {\mathrm e}^{-\beta \varepsilon_r})^N$
$= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $
$= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (\varepsilon_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$
where $\varepsilon_r$ are the energy eigenvalues we obtain
| $ \Xi^{\pm}(\beta,\mu) = \prod_{r}\left(1-(\pm1)\ \mathrm e^{\beta\ (\varepsilon_r-\mu)}\right)^{-(\pm1)} $ |
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When computing the Grand potential, one applies the $\log$ and the product becomes a sum. The particle number expectation value $\langle\hat N\rangle = - \frac{\partial}{\partial\mu}\Omega$ for these systems are
| $ \langle \hat N\rangle^{\pm} = \sum_r \langle \hat n_r\rangle^{\pm} $ |
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with the partition into energy levels given by the Einstein-Bose resp. Fermi-Dirac distribution functions
| $ \langle \hat n_r\rangle^{\pm} \propto \frac{1}{\mathrm e^{\beta\ (\varepsilon_r-\mu)}-(\pm 1)} $ |
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The $\propto$ expresses takes into account the possible multiplicity of states $r,r',\cdot$ with equal energy $\varepsilon_r,\varepsilon_{r'},\dots$.
The above derivation presupposes that the different energy values are given by the discretely indexed expressions of the form $\sim N \varepsilon$. This is the historical beginning of quantum mechaniscs.
Wikipedia: Partition function (statistical mechanics)