TABLE I: Specific heat C$_{q}$ and the Tsallis entropy S$_{q}$ of ideal gases [ S (1) 1 is given by Eqs. (44), q$_{U}$ = 1 + 2 /DN and q$_{L}$ = 1 − 1 / ( DN/ 2 + 1) = 1 /q$_{U}$ . In the superstatistics [36, 37], the χ $^{2}$- distribution with rank n is employed: G ( n ) = ψ ( n/ 2) − ln( n/ 2) ∝ 1 /n for n ≫ 1. NA : not applicable] method range C$_{q}$ S$_{q}$ (for | q − 1 | ≪ 1 . 0) S$_{q}$ (for N | q − 1 | ≫ 1) q -average 0 < q < q$_{U}$ k$_{B}$DN 2 NS (1) 1 − ( q − 1) [ N 2 2 ( S (1) 1 ) 2 − DN 4 ] k$_{B}$N 2 e $^{(1}$− q ) NS (1) 1 /k$_{B}$ normal average q$_{L}$ < q < ∞ k$_{B}$DN 2 NS (1) 1 − ( q − 1) [ N 2 2 ( S (1) 1 ) 2 + DN 4 ] k$_{B}$ ( $^{q}$− 1) Superstatistics n > 2 k$_{B}$DNn 2( $^{n}$− 2) NA N [ S (1) 1 − k$_{B}$D 2 G ( n ) ] (v) the entropy in first-principles approaches to the superstatistics of Refs. [36, 37] is additive ( S ( N ) = NS $^{(1)}$) while the Tsallis entropy is nonadditive. Items (i), (iii) and (iv) are consistent with results obtained with the q -Gaussian PDF [47]. The item (ii) shows that the rule of energy equipartition holds in both BG and nonextensive statistics. The item (iii) implies that the calculated properties of the entropy considerably depend on the adopted average methods [47]. It would be necessary to further examine q - and normal averages from various viewpoints [26]-[29] and to develop a first-principles superstatistical method which yields an entropy in conformity with the Tsallis one. This work is partly supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology of Japan. Acknowledgments 19