content_image/361050.jpg content_image/361051.jpg content_image/361052.jpg content_image/361053.jpg content_image/361054.jpg content_image/361055.jpg content_image/361056.jpg content_image/361057.jpg content_image/361058.jpg
U$_{q}$ in Eq. (19) is expressed in terms of α (see Appendix B1), 

<img src='content_image/361050.jpg'>

From Eqs. (18) and (61), U$_{q}$ and α are self-consistently determined as 

<img src='content_image/361051.jpg'>

<img src='content_image/361052.jpg'>

Equation (62) yields the specific heat given by 

<img src='content_image/361053.jpg'>

The specific heat of harmonic oscillators was discussed in Ref. [19] with the use of un- normalized [5] and normalized MEMs [2]. Their expressions for the internal energy U$_{q}$ have rather complicated q dependence [see Eqs. (12) and (22) of Ref. [19]]. Equation (64) expresses the Dulong-Petit specific heat: C$_{BG}$ = k$_{B}$DN in the BG statistics, which is in agreement with Ref. [18]. 

The Tsallis entropy is given by Eq. (40) with 

The BG entropy in the limit of q = 1 . 0 is given by 

with 

## 2. Normal average 

<img src='content_image/361054.jpg'>

<img src='content_image/361055.jpg'>

<img src='content_image/361056.jpg'>

<img src='content_image/361057.jpg'>

For harmonic oscillators under consideration, we obtain (see Appendix B2) 

<img src='content_image/361058.jpg'>

