===== Grand canonical partition function =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $J\in\mathbb N$ |
| $j\in \{1,\dots,J\}$ |
| @#DDDDDD: range       | @#DDDDDD: $N_j\in \mathbb N$ |
| @#55CCEE: context     | @#55CCEE: $N_j^\text{max}\in \mathbb N \cup \{\infty\}$ |
| @#55CCEE: context     | @#55CCEE: $Z_{N_j}(\beta) $ sequences in $N_j$ of canonical partition functions with length $N_j^\text{max}$ |
| @#FFBB00: definiendum | @#FFBB00: $\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.

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=== Discussion ===
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.
=== Theorems ===
== Bose-Einstein and Fermi-Dirac Statistics ==
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)} $ ^

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} $ ^

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)} $ ^ 

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. 

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=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Grand_canonical_partition_function|Partition function (statistical mechanics)]]
=== Context ===
[[Classical canonical partition function]], [[Quantum canonical partition function]], [[Fugacity]]