\cdot \hat W^{(2)} \cdot  \hat W, \quad {\rm etc.}\nonumber
 \end{aligned}$$

Finally, le us state an important result due to Pomeau [@pomeau82]. The $K$-time correlation functions imposes some restrictions on the $W$-matrix.

Because of the time reversal symmetry, for all indices $i_1,i_2,\cdots i_K = \{ 1,2,\cdots K\}$, the symmetry relation $$w_{i_1i_2} w_{i_2i_3} \cdots w_{i_{K-1}i_K} w_{i_Ki_1} = w_{i_1i_K} w_{i_K i_{K-1}}   \cdots w_{i_3i_2}  w_{i_2i_1}\label{pomocondition}$$ must be satisfied.

In the following we apply this coarse graining approach to compute the probability transfer matrix for some particular cases.

Specific computation of the probability transition matrix in various situations. {#Results}
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In this section we shall apply the coarse graining approach to the Q2R dynamics in the case of small lattice size. In Ref. [@chine] we have fairly explored the computation of the probability transition matrix, in particular, in the case of extended systems ($N=256\times 256$), however in this case the cycles are usually huge, therefore this general approach is not really complete. In this sense, we focus our effort in treating moderate system sizes, namely $N=4\times 4$, $N=8\times 8$, $N=16\times 16$, having all of them tractable cycles.

The robustness of the methodology.
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In general for a system of small size, one is able to 