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 $$\begin{equation*} \frac{d w_i(t)}{dt} = 0. \end{equation*}$$