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 $$ \begin{equation*} Z=\sum_j \exp{(-\beta(E_i))}. \end{equation*} $$