content_image/371517.jpg content_image/371515.jpg content_image/371516.jpg content_image/371513.jpg content_image/371514.jpg content_image/371518.jpg
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 

<img src='content_image/371517.jpg'>

<img src='content_image/371515.jpg'>

<img src='content_image/371516.jpg'>

<img src='content_image/371513.jpg'>

<img src='content_image/371514.jpg'>

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 

<img src='content_image/371518.jpg'>

which is quite different from S$_{q}$ in Eq. (47) obtained by the q -average. 

## C. Model calculations 

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 

