content_image/354573.jpg content_image/354574.jpg content_image/354576.jpg content_image/354575.jpg content_image/354572.jpg content_image/354577.jpg content_image/354578.jpg
which leads to 

Equation (23) may be rewritten as 

with 

˜ U$_{q}$ is expressed by 

<img src='content_image/354573.jpg'>

<img src='content_image/354574.jpg'>

<img src='content_image/354576.jpg'>

<img src='content_image/354575.jpg'>

<img src='content_image/354572.jpg'>

Calculation methods discussed in this section will be applied to classical ideal gases and harmonic oscillators in Secs. III and IV, respectively. 

## III. IDEAL GASES 

## A. q -average 

We consider hamiltonian for N -body D -dimensional ideal gases given by 

<img src='content_image/354577.jpg'>

where m stands for the mass and p i momentum of ideal gases. By using the exact method [13, 48], we obtain the partition function Z$_{q}$ in Eq. (17) (for details, see Appendix B1), 

<img src='content_image/354578.jpg'>

