content_image/345458.jpg content_image/345459.jpg content_image/345460.jpg content_image/345461.jpg content_image/345463.jpg content_image/345464.jpg content_image/345465.jpg content_image/345466.jpg
By using the formula: 

we obtain S$_{q}$ for | q − 1 | ≪ 1 . 0 given by 

<img src='content_image/345458.jpg'>

<img src='content_image/345459.jpg'>

where the ( q − 1) term includes the O ( N $^{2}$) contribution showing the nonadditivity. 

The Tsallis entropy for large N (1 − q ) is given by 

<img src='content_image/345460.jpg'>

where we adopt the relation: ln Γ( z ) ≃ ( z − 1 / 2) ln z − z + (1 / 2) ln 2 π + ·· for | z | → ∞ . 

## B. Normal average 

For ideal gases whose hamiltonian is given by Eq. (33), we obtain (for details, see Appendix B2) 

˜ U$_{q}$ in Eq. (32) may be expressed in terms of ˜ α (see Appendix B2), 

<img src='content_image/345461.jpg'>

<img src='content_image/345463.jpg'>

From Eqs. (31) and (49), ˜ U$_{q}$ and ˜ α are self-consistently determined as 

A conceivable q value is given by 

<img src='content_image/345464.jpg'>

<img src='content_image/345465.jpg'>

<img src='content_image/345466.jpg'>

