In statistical physics this condition ensures that it is e.g., the Boltzmann distribution which is generated when equilibrium is reached.

We introduce now the Boltzmann distribution $$\begin{equation*} w_i= \frac{\exp{(-\beta(E_i))}}{Z}, \end{equation*}$$ which states that the probability of finding the system in a state $i$ with energy $E_i$ at an inverse temperature $\beta = 1/k_BT$ is $w_i\propto \exp{(-\beta(E_i))}$. The denominator $Z$ is a normalization constant which ensures that the sum of all probabilities is normalized to one. It is defined as the sum of probabilities over all microstates $j$ of the system $$\begin{equation*} Z=\sum_j \exp{(-\beta(E_i))}. \end{equation*}$$