We may express ˜ U$_{q}$ given by Eq. (32) in terms of ˜ α as given by (see Appendix B2) From Eqs. (31) and (70), ˜ U$_{q}$ and ˜ α are self-consistently determined as where q$_{U}$ is given by Eq. (63). We obtain the specific heat given by Calculated specific heat is the same as the Boltzmann-Gibbs result of C$_{BG}$ = k$_{B}$DN . The Tsallis entropy may be calculated by Eq. (54) with where ν$_{q}$ is given by Eq. (66). The properties of the specific heat and Tsallis entropies of harmonic oscillators are es- sentially the same as those of ideal gases if we read DN/ 2 → DN and m → 1 /ω . It is interesting to examine the fluctuation of energy in nonextensive systems, which has been studied in Refs. [18, 44, 45]. The q -average of H 2 is given by (see Appendix B1) where η = 1 / 2 and η = 1 for ideal gases and harmonic oscillators, respectively. The relative fluctuation of the energy is given by A. Fluctuation of the energy V. DISCUSSION ˜ U$_{q}$ = DN [ DN ( q − 1) + q ]˜ α . (70) ˜ U$_{q}$ = DN β = k$_{B}$ DNT, (71) ˜ α = β q + DN ( q − 1) for q > 1 − 1 ( DN +1) ≡ q$_{L}$ = 1 /q$_{U}$, (72) ˜ C$_{q}$ = k$_{B}$DN. (73) ˜ c$_{q}$ = ( ˜ Z$_{q}$ ) $^{1}$− q ˜ ν$_{q}$ , (74) ˜ ν$_{q}$ = DN ( 1 − 1 q ) + 1 = ν$_{1}$$_{/q}$, (75) [ H $^{2}$]$_{q}$ = ηDN ( ηDN + 1) [1 + ηDN (1 − q )] β $^{2}$[1 + ( ηDN + 1)(1 − q )] , (76) U$_{q}$ √ ηDN [1 + ( ηDN + 1)(1 − q )] ∝    1 √$_{DN}$ for q = 1 . 0 , 1 DN for DN (1 − q ) ≫ 1 . (78) 13 Δ U$_{q}$ U$_{q}$ ≡ √ [ H 2 − U 2 $_{q}$]$_{q}$ U$_{q}$ = 1 , (77)