which leads to Equation (23) may be rewritten as with ˜ U$_{q}$ is expressed by Calculation methods discussed in this section will be applied to classical ideal gases and harmonic oscillators in Secs. III and IV, respectively. We consider hamiltonian for N -body D -dimensional ideal gases given by 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), III. IDEAL GASES A. q -average ˜ γ = β q . (28) ˜ p$_{q}$ = 1 ˜ Z$_{q}$ [ 1 − ( q − 1 q ) ˜ α H ] 1 / ( $^{q}$− 1) + , (29) ˜ Z$_{q}$ = Tr [ 1 − ( q − 1 q ) ˜ α H ] 1 / ( $^{q}$− 1) + , (30) ˜ α = β/q 1 − (1 /q − 1) β ˜ U$_{q}$ . (31) ˜ U$_{q}$ = 1 ˜ Z$_{q}$ Tr { [ 1 − ( q − 1 q ) ˜ α H ] 1 / ( $^{q}$− 1) + H } . (32) H = N ∑ i =1 H$_{i}$ = N ∑ i =1 p 2 i 2 m , (33) Z$_{q}$ =      V N N ! h DN [ 2 π m ( $^{q}$− 1) α ] DN/ 2 Γ ( 1 q$_{−}$ 1 − DN 2 ) Γ ( 1 q$_{−}$ $_{1}$) 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 $_{$_{−}$q}$+1 ) Γ ( 1 1 $_{$_{−}$q}$+ DN 2 +1 ) for q < 1 , (34) 7