By using the formula: we obtain S$_{q}$ for | q − 1 | ≪ 1 . 0 given by 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 where we adopt the relation: ln Γ( z ) ≃ ( z − 1 / 2) ln z − z + (1 / 2) ln 2 π + ·· for | z | → ∞ . For ideal gases whose hamiltonian is given by Eq. (33), we obtain (for details, see Appendix B2) From Eqs. (31) and (49), ˜ U$_{q}$ and ˜ α are self-consistently determined as A conceivable q value is given by ln ( Γ( z + a ) Γ( z ) z a ) ≃ − a (1 − a ) 2 z for | z | → ∞ , (45) S$_{q}$ = NS (1) 1 − ( q − 1) [ N 2 2 ( S (1) 1 ) 2 − DN 4 ] + ·· , (46) S$_{q}$ ≃ ( k$_{B}$N 2 ) e $^{(1}$− q ) NS (1) 1 /k$_{B}$ for N (1 − q ) ≫ 1 , (47) ˜ Z$_{q}$ =      V N N ! h DN [ 2 π m ( $^{q}$− 1)˜ α ] DN/ 2 Γ ( 1 q$_{−}$ $_{1}$+1 ) Γ ( 1 q$_{−}$ $_{1}$+ DN 2 +1 ) for 1 < q < 3 , V N N ! h DN ( 2 π m ˜ α ) DN/ 2 for q = 1 , V N N ! h DN [ 2 πm $^{(1}$− q )˜ α ] DN/ 2 Γ ( 1 1 $_{−}$q − DN 2 ) Γ ( 1 1 $_{$_{−}$q}$) for q < 1 . (48) ˜ U$_{q}$ in Eq. (32) may be expressed in terms of ˜ α (see Appendix B2), ˜ U$_{q}$ = DN [ DN ( q − 1) + 2 q ]˜ α . (49) ˜ U$_{q}$ = DN 2 β = k$_{B}$DNT 2 , (50) ˜ α = β q + DN 2 ( q − 1) . (51) q > 1 − 1 ( DN 2 + 1 ) ≡ q$_{L}$ = 1 q$_{U}$ , (52) 9 B. Normal average