find some cycles for a given energy. Building a time series for the magnetization $\{ M(t)\}= \{ M_1,M_2,\cdots M_T\}$, then one defines a partition on the possible values of the magnetization, as explained in a previous Sec. \[masterEqn\]. In the cases considered here, it is always possible to use the finest possible partition, that is, for the exact available values of the magnetization (something impractical in large systems). In this case the partitions are composed by a set of $N+1$ ($N$ is assumed to be even) well defined values $ M=\{-N, -N+2,-N+4,\cdots , N-2,N\}$. That is for $4\times 4$ the partition has a maximum of 17 elements, for $N=8\times 8$ there are 65 elements, and for $N=16\times 16$ the partition possesses a maximum of 257 elements.

A first result concerns the equivalence of the probability density function of magnetization obtained via the time series of the magnetization and the equilibrium distribution resulting from the eigenvectors of the probability transition matrix $\hat W$. Hence, the results arising from temporal averages and transition probability matrix in the configurational space are consistent themselves. This fact ensures a first validation of the method. However, the probability transition matrix provides extra information of a system, 