notices that all partitions produce similar results. ](Feq_8x8_E_0_Diff_W.eps "fig:"){width="7cm"}

\(b) ![ \[Method88\] (a) Plot of the equilibrium distribution $f_{eq}$ vs $M$ for a $8\times 8$ system with $E=0$ and a cycle of a period $T=672018$ for three different partitions of the magnetization values. The plot shows that all distribution functions lies under the same curve. The inset shows the parabolic behavior in magnetization which after a fit reads $ \log f_{eq}= -M^2/ 116$. (b) Plot of the second eigenmode ${\bm \chi}^{(2)}$ corresponding to the eigenvalue closest to the unit circle. One notices that all partitions produce similar results. ](Feq_2_8x8.eps "fig:"){width="7cm"}

In the following we summarize the methodology for the cases of $4\times 4$, $8\times 8$, and $16\times 16$, in all cases the full cycles were considered, and we provide the finest partition.

Exact calculation for various lattices.
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We have studied in detail the case of a $4\times 4$ periodic lattice, because the phase space possesses $2^{32} \approx 4\times10^9$ distinct configurations and the calculations are exactly realizable in a computer up to end, therefore it shows explicitly the method. It is shown that the coarse graining approach is fully applicable in the $4\times 4$ lattice case. We used different partition 