The last equation is very similar to the so-called Master equation, which relates the temporal dependence of a PDF \( w_i(t) \) to various transition rates. The equation can be derived from the so-called Chapman-Einstein-Enskog-Kolmogorov equation. The equation is given as $$ \begin{equation} \tag{4} \frac{d w_i(t)}{dt} = \sum_j\left[ W(j\rightarrow i)w_j-W(i\rightarrow j)w_i\right], \end{equation} $$ which simply states that the rate at which the systems moves from a state \( j \) to a final state \( i \) (the first term on the right-hand side of the last equation) is balanced by the rate at which the system undergoes transitions from the state \( i \) to a state \( j \) (the second term). If we have reached the so-called steady state, then the temporal development is zero. This means that in equilibrium we have $$ \begin{equation*} \frac{d w_i(t)}{dt} = 0. \end{equation*} $$