Specific heat and entropy of $N$-body nonextensive systems

We have studied finite $N$-body $D$-dimensional nonextensive ideal gases and harmonic oscillators, by using the maximum-entropy methods with the $q$- and normal averages ($q$: the entropic index). The validity range, specific heat and Tsallis entropy obtained by the two average methods are compared. Validity ranges of the $q$- and normal averages are $0 < q < q_U$ and $q > q_L$, respectively, where $q_U=1+(\eta DN)^{-1}$, $q_L=1-(\eta DN+1)^{-1}$ and $\eta=1/2$ ($\eta=1$) for ideal gases (harmonic oscillators). The energy and specific heat in the $q$- and normal averages coincide with those in the Boltzmann-Gibbs statistics, % independently of $q$, although this coincidence does not hold for the fluctuation of energy. The Tsallis entropy for $N |q-1| \gg 1$ obtained by the $q$-average is quite different from that derived by the normal average, despite a fairly good agreement of the two results for $|q-1 | \ll 1$. It has been pointed out that first-principles approaches previously proposed in the superstatistics yield $additive$ $N$-body entropy ($S^{(N)}= N S^{(1)}$) which is in contrast with the $nonadditive$ Tsallis entropy.