description using gross or macroscopic variables, namely the total magnetization of the system (\[magnetization\]), instead of microscopic variables. Let us denote by $ \rho_t(M)$, the reduced probability distribution in terms of the variable $M$. The final master equation will be of the form $\rho_{t+1}(M) = {\mathcal W}  \rho_{t}(M).  $ The linear operator, $ {\mathcal W}$, acts only in the subspace of constant $E$, but is spanned over arbitrary values of the magnetization.

As the original Perron-Frobenius equation, $ {\mathcal W}$ depends explicitly of the Q2R rule and it may be computed in principle. However, in practice, it is not possible because of the complex structure of the dynamics, in particular because of the existence of a myriad of different periods for a given $E$. Empirically, it seems that the probability to reach a long period is finite, therefore it is probable that any random initial condition would reach a extremely long cycle and the sequence of magnetizations would represent a Markov chain $\{\cdots,M_{t-1},M_t,M_{t+1},\cdots\}$. Moreover, it is possible to reduce again the information via a coarse-graining partition of the possible values of the Markov chain. This partition is defined through a finite number of sets of no overlapping intervals: $I_1 =[ 