with Equations (91) and (93) are similar to Eqs. (16) and (29), respectively, although the ranges of conceivable q values are not necessarily the same. 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 from which we obtain ψ ( x ) standing for the poli-gamma function. Similarly from Eqs. (62) and (68) for harmonic oscillators, we obtain and then ˜ q = 1 − 2 DN + n . (94) 2. Energy and entropy E ( β ) = DN 2 β , (95) S ( β ) = k$_{B}$N [( D 2 ) ln ( 2 πm h $^{2}$β ) + ln σ + ( D 2 + 1 )] , (96) E = DN 2 ¯ β ( n n − 2 ) , (97) S = k$_{B}$N [( D 2 ) ln ( 2 πm h 2 ¯ β ) + ln σ + ( D 2 + 1 )] − k$_{B}$DN 2 [ ψ ( n 2 ) − ln ( n 2 )] , (98) E ( β ) = DN β , (99) S ( β ) = k$_{B}$DN [ ln ( 2 π hωβ ) + 1 ] , (100) E = DN ¯ β ( n n − 2 ) , (101) S = k$_{B}$DN [ ln ( 2 π hω ¯ β ) + 1 ] − k$_{B}$DN [ ψ ( n 2 ) − ln ( n 2 )] . (102) 16