sequence, $\{ M(t)\}$, but for four different lengths of the time series: $T= 43115258$ (the complete cycle), and $T^* = 10^6 ,\, 5\times 10^6,\, \& \,  20\times 10^6$ (partial sequences). Visually one sees that there is not substantial difference among the different values of $T^*$. Moreover, the following Table \[TableTest\] compares quantitatively the mean square difference measuring $Q_1 = || {\bm f}^{*} -{\bm f}_{eq} ||^2/K$ and $Q_2 =\sum_{i=1}^K|\lambda_i -\lambda_i^*|^2/K$. Here $K$ is the number of partitions.

\(a) ![ \[Method\] (a) Plot of the equilibrium distribution ${\bm f}_{eq} $ for the case of a $16\times 16$ system with $E=-292$ ($E/N \approx 1.14$) and a cycle of a period $T=43115258$. The computation of ${\bm f}_{eq} $ is compared with shorter sequences of the same time series of length $T^* = 10^6 ,\, 5\times 10^6,\, \& \,  20\times 10^6$. (b) The set of 257 eigenvalues of the $\hat W$-matrix for the same conditions of (b). Naturally, $\lambda =1$ is present by the definition of the probability transfer matrix $\hat W$. ](feq_N_16x16_E_292_T.eps "fig:"){width="7cm"}

\(b) ![ \[Method\] (a) Plot of the equilibrium distribution ${\bm f}_{eq} $ for the case of a $16\times 16$ system with $E=-292$ ($E/N \approx 1.14$) and a cycle of a period 