content_image/356315.jpg content_image/356316.jpg content_image/356317.jpg content_image/356318.jpg content_image/356319.jpg content_image/356320.jpg content_image/356323.jpg content_image/356321.jpg content_image/356322.jpg
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 

<img src='content_image/356315.jpg'>

<img src='content_image/356316.jpg'>

where q$_{U}$ is given by Eq. (63). We obtain the specific heat given by 

<img src='content_image/356317.jpg'>

<img src='content_image/356318.jpg'>

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). 

<img src='content_image/356319.jpg'>

<img src='content_image/356320.jpg'>

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 . 

## V. DISCUSSION 

## A. Fluctuation of the energy 

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 

<img src='content_image/356323.jpg'>

<img src='content_image/356321.jpg'>

<img src='content_image/356322.jpg'>

