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tags: stat mech, lecture
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# L06 Internal and total chemical potential. Grand canonical ensemble

## Reading
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## Notes
[class notes](https://web.physics.utah.edu/~rogachev/7310/L06%20Grand%20canonical.pdf)

### Grand Canonical Ensembles
T is fixed. N, E can be exchanged between system and reservoir.
Reservoir is huge, so chemical potential $\mu$ stays the same

#### Grand Partition Function
$Z_G = \sum e^{-\beta E_{x_i} + \beta \mu N_{x_i}}$
$\hspace{.8cm} = \sum_{N_j}e^{+\beta \mu (N_j)}*\sum_{x_i(N_j)}e^{+\beta E_{x_i}(N_j)}$
$\hspace{.8cm} = \sum_{N_j}e^{+\beta \mu (N_j)}*Z(N_j,V_j,T)$

$\hspace{2cm} F = -kTln(Z)$

$\hspace{.8cm} = e^{\beta \mu N^*}e^{-\beta F}$
$\hspace{.8cm} = e^{-\beta(F-\mu N)}$

Then probability of a particular microstate:
$P_{x_i} = \frac{e^{-\beta E_{x_i} + \beta \mu N_{x_i}}}{Z_G}$