Partition Function
#Physics
$\displaystyle Z=\sum_{s}e^{- \beta\varepsilon_{s}}$
- Dimensionless probability normalization factor (the below equation has to sum up to one when evaluating over all energies
- We sum over all of the microstates with index $\displaystyle s$
- $\displaystyle e^{- \beta\varepsilon}$ is the Boltzmann factor
$\displaystyle Z_{1}=n_{Q}V$
- This is the partition function for one particle in a certain volume
- $\displaystyle n_{Q}$ is the [quantum concentration]
- $\displaystyle V$ is the volume the gas occupies (can be area or length if 2D or 1D)
$\displaystyle Z_{N \text{distinguishable}}=Z_{1}^{N}$
- Example is a spin magnet system in classical mechanics
$\displaystyle Z_{N \text{indistinguishable}}=\frac{Z_{1}^{N}}{N!}$
- Example is an ideal gas
- $\displaystyle Z_{1}$ is the partition function for one particle
Applications
$\displaystyle N=\frac{1}{\beta}\frac{ \partial }{ \partial \mu }\ln Z$
- $\displaystyle N$ is the number of particles
- $\displaystyle \mu$ is the [chemical potential]
$\displaystyle E_{\text{tot}}-\mu N=-\frac{ \partial }{ \partial \beta }\ln Z$
- $\displaystyle E_{\text{tot}}$ is the energy of the system + environment