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We have discussed the Tsallis entropy in finite $N$-unit nonextensivesystems, by using the multivariate $q$-Gaussian probability distributionfunctions (PDFs) derived by the maximum entropy methods with the normal averageand the $q$-average ($q$: the entropic index). The Tsallis entropy obtained bythe $q$-average has an exponential $N$ dependence: $S_q^{(N)}/N \simeq\:e^{(1-q)N \:S_1^{(1)}} $ for large $N$ ($\gg \frac{1}{(1-q)} >0$). Incontrast, 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 aregenerally quite different, although the both results are in fairly goodagreement for $\vert q-1 \vert \ll 1.0$. The validity of the factorizationapproximation to PDFs which has been commonly adopted in the literature, hasbeen examined. We have calculated correlations defined by $C_m= \langle (\deltax_i \:\delta x_j)^m \rangle -\langle (\delta x_i)^m \rangle\: \langle (\deltax_j)^m \rangle$ for $i \neq j$ where $\delta x_i=x_i -\langle x_i \rangle$, andthe bracket $\langle \cdot \rangle$ stands for the normal and $q$-averages. Thefirst-order correlation ($m=1$) expresses the intrinsic correlation andhigher-order correlations with $m \geq 2$ include nonextensivity-inducedcorrelation, whose physical origin is elucidated in the superstatistics.

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