Boltzmann Distribution

Weight

For a system in thermal contact with a reservoir at temperature $T$, the probability of a microstate with energy $E_i$ is

$$p_i=\frac{e^{-\beta E_i}}{Z}, \qquad \beta=\frac{1}{k_BT}.$$

Partition Function

The normalization constant

$$Z=\sum_i e^{-\beta E_i}$$

is the partition function. It is more than normalization: derivatives of $\ln Z$ give thermodynamic quantities.