The entropy in finite $N$-unit nonextensive systems: the ordinary average and $q$-average

We have discussed the Tsallis entropy in finite $N$-unit nonextensive systems, by using the multivariate $q$-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the $q$-average ($q$: the entropic index). The Tsallis entropy obtained by the $q$-average has an exponential $N$ dependence: $S_q^{(N)}/N \simeq \:e^{(1-q)N \:S_1^{(1)}}$ for large $N$ ($\gg \frac{1}{(1-q)} >0$). In contrast, the Tsallis entropy obtained by the normal average is given by $S_q^{(N)}/N \simeq [1/(q-1)N]$ for large $N$ ($\gg \frac{1}{(q-1)} > 0)$. $N$ dependences of the Tsallis entropy obtained by the $q$- and normal averages are generally quite different, although the both results are in fairly good agreement for $\vert q-1 \vert \ll 1.0$. The validity of the factorization approximation to PDFs which has been commonly adopted in the literature, has been examined. We have calculated correlations defined by $C_m= \langle (\delta x_i \:\delta x_j)^m \rangle -\langle (\delta x_i)^m \rangle\: \langle (\delta x_j)^m \rangle$ for $i \neq j$ where $\delta x_i=x_i -\langle x_i \rangle$, and the bracket $\langle \cdot \rangle$ stands for the normal and $q$-averages. The first-order correlation ($m=1$) expresses the intrinsic correlation and higher-order correlations with $m \geq 2$ include nonextensivity-induced correlation, whose physical origin is elucidated in the superstatistics.