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tags: stat mech, lecture
---

# L04 Canonical ensemble

## Readings ([reference material](/fy9b5mX0Rui-Sii3wHTncg))
- Reading Kittel Ch. 3
- Reading Sethna Ch 6.1, 6.2

## Notes
[**Lecture 03 class notes**](https://web.physics.utah.edu/~rogachev/7310/L04%20Canonical%20ensemble.pdf)

### Ensembles
- A [statistical ensemble](/JUkPDnqjQPmxuCONGrn4yw) is a collection of all accessible microstates of a system with associated probabilities.
- [Canonical ensembles](/JUkPDnqjQPmxuCONGrn4yw) are the collection of possible states of a mechanical system in thermal equilibrium with a heat bath (or reservoir) at a fixed temperature T
- A [microcanonical ensemble](/JUkPDnqjQPmxuCONGrn4yw) is a completely closed/isolated system


#### Helmholtz free energy
$F = TS - E$
$\hspace{.5cm} = -kTln(Z)$

$dF = -SdT - PdV$

From which we have:

${\displaystyle S=\left.-\left({\frac {\partial F}{\partial T}}\right)\right|_{V,N},\quad P=\left.-\left({\frac {\partial F}{\partial V}}\right)\right|_{T,N},\quad \mu =\left.\left({\frac {\partial F}{\partial N}}\right)\right|_{T,V}.}$

#### Ideal gas partition function
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Statistical_Mechanics/Boltzmann_Average/Ideal_Gas_Partition_Function

### Partition function to get functions of state
- [Using the partition function to obtain functions of state](https://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/A1/2011/handout6.pdf)