---
tags: stat mech, lecture
---

# L05 Gibbs canonical ensemble

## Reading
"No good reading" lol :pray:
[This](https://scholar.harvard.edu/files/schwartz/files/7-ensembles.pdf) might be helpful though. Check out section 5.

## Notes

### Generalized force, displacement, and work
Using the 1st law of thermo:

$dE = TdS - PdV + \mu dN$
$\hspace{.8cm} = heat + work$
$\hspace{.8cm} = dQ + dW$

<!-- Note that E is a [state function](/@OptXFinite-PhD-Notes/statmech-dictionary), so the quantity dE is independent of path
-->

Now $dW$ depends on the parameters we use for force and displacement
| System | Generalized Force ($\mathcal{J}$) | Generalized Displacement ($x$) | dW |
| ---: | :--- | :--- | :---: |
| 3-D | $P$ (pressure) | $V$ (volume) | $-PdV$ |
| 2-D |$\sigma$ (surface tension) | $A$ (area) | $\sigma$dA |
| 1-D | $f$ (force) | $l$ (length) | $fdl$ |
| magnetic materials| $H$ (auxiliary field)| $M$ ([magnetization](/@OptXFinite-PhD-Notes/statmech-dictionary))|$HdM$|
| electrical | $E$ (electric field) | $P$ (polarization) | $EdP$ |

### The Gibbs ensemble
> In both the microcanonical and canonical ensembles, we fix the volume. We could instead let the volume vary and sum over possible volumes. Allowing the volume to vary gives the Gibbs ensemble. In the Gibbs ensemble, the partition function depends on pressure rather than volume, just as the canonical ensemble depended on temperature rather than energy.
>
> In the microcanonical, canonical, and Gibbs ensembles, the number of particles N in the system is fixed.

Gibbs ensemble involves an external generalized force acting on a system. System has N and T fixed, but the generalized displacement can vary.

#### Gibbs Partition Function
$Z_j = \sum e^{-\beta E_{x_i} + \beta \mathcal{J}x}$

#### Gibbs Potential (Free Energy)
remember thermodynamic potential $\rightarrow$ energy

$G = -kT ln(Z_j)$

$\hspace{0.5cm} = E - TS + PV$