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Grand canonical partition function
Definition
| $ (Z_N(\beta))_N $ … sequence of canonical partition functions |
| $\Xi(\beta,\mu):=\sum_{N=0}^\infty z(\beta,\mu)^N\cdot Z_N(\beta) $ |
Here $z$ denotes the fugacity.
Discussion
This mirrors the classical microcanonical phase volume and the classical canonical partition function.
The summands $z(\beta,\mu)^N\cdot Z_N(\beta)$ can be viewed as canonical partition function where the distributions $\mathrm e^{-\beta\ H_N}$ are shifted to $\mathrm e^{-\beta\ (H_N-\mu\ N)}$. Accordingly the meaning of the so called chemical potential $\mu$ is a package of energy associated with each given particle in the system.
Theorems
Important grand canonical partition functions in QM are those for bosons and fermions, denoted $\Xi^+$ and $\Xi^-$, respectively. The derivation uses $\sum_N\mathrm e^{-\beta\sum_r N\ h_r}=\prod_r \sum_N (e^{-\beta\ h_r})^N$ and so they turn out to be
| $ \Xi^{\pm}(\beta,\mu) = \prod_{r}\left(1-(\pm1)\ \mathrm e^{\beta\ (\varepsilon_r-\mu)}\right)^{-(\pm1)} $ |
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where $\varepsilon_k$ are the energy eigenvalues. The discussion in the entry density of states relates to this.
The particle number expectation values for these systems are
| $ \langle \hat N\rangle^{\pm} = \sum_r \langle \hat n\rangle^{\pm} $ |
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with the partition into energy levels given by the Einstein-Bose resp. Fermi-Diract distribution functions:
| $ \langle \hat n_r\rangle^{\pm} = \left(\mathrm e^{\beta\ (\varepsilon_r-\mu)}-(\pm 1)\right)^{-1} $ |
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Parents
Reference
Wikipedia: Partition function (statistical mechanics)
Requirements
