Boltzmann Distribution

Definition

Boltzmann distribution models the probability that a system is in a specific state as a function of that state’s energy and temperature of the system.

$p(s)=\frac{1}{Z}\exp \left(-\frac{E(s)}{T}\right)$ where $E(s)$ is the energy of the state $s$, $T$ is the temperature of the system, $Z=\sum _{s\in S}\exp \left(-\frac{E(s)}{T}\right)$ is the normalizing constant

The distribution shows that states with lower energy always have a higher probability. Higher temperature makes the distribution flat.