content_image/333295.jpg content_image/333297.jpg content_image/333298.jpg content_image/333299.jpg content_image/333300.jpg content_image/333303.jpg content_image/333304.jpg content_image/333301.jpg content_image/333302.jpg
with 

<img src='content_image/333295.jpg'>

Equations (91) and (93) are similar to Eqs. (16) and (29), respectively, although the ranges of conceivable q values are not necessarily the same. 

## 2. Energy and entropy 

The χ $^{2}$-distribution in Eq. (83) is given ad hoc [30]-[33]. Several first-principles ap- proaches have been proposed to determine the optimum distribution of β in the su- perstatistics [34]-[37]. Ref.[37] considered the local energy E ( β ) = 〈 H 〉 Π and entropy S ( β ) = − 〈 ln Π 〉 Π for local equilibrium states with an inverse temperature β , and then obtained the energy and entropy of the system given by E = 〈 E ( β ) 〉 β and S = 〈 S ( β ) 〉 $_{β}$, where 〈·〉 Π and 〈·〉 β express averages over Π( β ) and f ( β ), respectively. Equations (36) and (44) for ideal gases lead to 

<img src='content_image/333297.jpg'>

<img src='content_image/333298.jpg'>

from which we obtain 

<img src='content_image/333299.jpg'>

ψ ( x ) standing for the poli-gamma function. Similarly from Eqs. (62) and (68) for harmonic oscillators, we obtain 

and then 

<img src='content_image/333300.jpg'>

<img src='content_image/333303.jpg'>

<img src='content_image/333304.jpg'>

<img src='content_image/333301.jpg'>

<img src='content_image/333302.jpg'>

