Partition Function

$Z=\sum _s{e}^{-\beta {\epsilon }_s}$

• 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 $s$
• $e^{-\beta \epsilon }$ is the Boltzmann factor

$Z_1=n_{Q}V$

• This is the partition function for one particle in a certain volume
• $n_Q$ is the [quantum concentration]
• $V$ is the volume the gas occupies (can be area or length if 2D or 1D)

$Z_{N\text{distinguishable}}=Z_1^N$

• Example is a spin magnet system in classical mechanics

$Z_{N\text{indistinguishable}}=\frac{Z_1^N}{N!}$

• Example is an ideal gas
• $Z_1$ is the partition function for one particle

Applications

$N=\frac{1}{\beta }\frac{\partial }{\partial \mu }\ln Z$

• $N$ is the number of particles
• $\mu$ is the [chemical potential]

$E_{\text{tot}}-\mu N=-\frac{\partial }{\partial \beta }\ln Z$

• $E_{\text{tot}}$ is the energy of the system + environment

$\Delta (U^2)=-\frac{{\partial }^2}{\partial {\beta }^2}\ln Z$