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.