where q$_{U}$ is given by Eq. (38). Equation (50) yields specific heat given by which is the same as that of the BG statistics. The Tsallis entropy is expressed by with where ν$_{q}$ is given by Eq. (42). For | q − 1 | ≪ 1 . 0, we obtain ˜ S$_{q}$ given by where S (1) 1 is given by Eq. (44). Equation (57) is the same as Eq. (46) except for the O ( DN ) term in the bracket. For large N ( q − 1), ˜ S$_{q}$ is given by which is quite different from S$_{q}$ in Eq. (47) obtained by the q -average. We will present some numerical calculations of the Tsallis entropy of ideal gases with D = 1. Equations (38) and (52) show that the conceivable q values for q - and normal averages are 0 < q < q$_{U}$ and q > q$_{L}$ , respectively, where q$_{U}$ = 1+2 /DN and q$_{L}$ = 1 − 2 / ( DN +2) (see the inset of Fig. 1). q$_{L}$ and q$_{U}$ approach unity for N → ∞ . Although the N dependence of the Tsallis entropy in the q -average is rather different from that in the normal average, as Eqs. (47) and (58) show, both results are in fairly good agreement for q$_{L}$ < q < q$_{U}$ where both the q - and normal averages are valid. Dashed and solid curves in Fig. 1 show C. Model calculations ˜ C$_{q}$ = k$_{B}$DN 2 , (53) ˜ S$_{q}$ = k$_{B}$ ( ˜ c$_{q}$ − 1 1 q ) , (54) ˜ c$_{q}$ = ( ˜ X$_{q}$ ) $^{1}$− q = ( ˜ Z$_{q}$ ) $^{1}$− q ˜ ν$_{q}$ , (55) ˜ ν$_{q}$ = DN 2 ( 1 − 1 q ) + 1 = ν$_{1}$$_{/q}$, (56) ˜ S$_{q}$ = NS (1) 1 − ( q − 1) [ N 2 2 ( S (1) 1 ) 2 + DN 4 ] + ·· , (57) ˜ S$_{q}$ ≃ k$_{B}$ q − 1 [ 1 − 2 q e − ( $^{q}$− 1) NS (1) 1 /k$_{B}$ N ( q − 1) ] for N ( q − 1) ≫ 1 , (58) 10