We wish to derive the required properties of the probabilities $T$ and $A$ such that $w_i^{(t\rightarrow \infty)} \rightarrow w_i$, starting from any distribution, will lead us to the correct distribution.

We can now derive the dynamical process towards equilibrium. To obtain this equation we note that after $t$ time steps the probability for being in a state $i$ is related to the probability of being in a state $j$ and performing a transition to the new state together with the probability of actually being in the state $i$ and making a move to any of the possible states $j$ from the previous time step.