# Partition Function

## $\LARGE Z=\sum_re^{\frac{-\varepsilon_r}{kT}}$

### The Terms

$Z$ - Partition function

$\varepsilon_r$ - Energy of the system when it is in microstate $r$

$k$ - Boltzmann constant [Value: 1.381 x 10-23 J K-1]

$T$ - Absolute temperature [Units: Kelvins, K]

### What Does It Mean?

The partition function is an important quantity in statistical mechanics which encodes the statistical properties of a system in thermodynamic equilibrium. This equation is particular to the canonical ensemble, that is the ensemble in which the system of interest is allowed to exchange heat with its surrounding environment, with a fixed temperature, volume and number of particles. There is also the grand canonical ensemble, in which the system can exchange heat and particles with the environment, with a fixed temperature, volume and chemical potential. The partition function for this ensemble is

$\Xi=\sum_r\sum_Ne^{\frac{-(\varepsilon_r-{\mu}N)}{kT}}.$

### Further information at Warwick

The partition function is covered in the second year module "PX265 Thermal Physics II" and third year module "PX366 Statistical Physics".