where we employ the definition of ˜ U$_{q}$ given by Equation (A8) leads to With the use of Eqs. (A10) and (54), the physical temperature is given by [38] leading to where we employ the relation: ∂ ˜ X$_{q}$/∂ ˜ U$_{q}$ = q ˜ γ ˜ X$_{q}$ . We first discuss evaluations by the q -average given by by using the exact expressions for the gamma function [13, 48]: Appendix: B. Evaluations of averages by the exact approach 1. The q -average ˜ U$_{q}$ = 1 ˜ X$_{q}$ Tr { [1 − ( q − 1)˜ γH ] 1 / ( $^{q}$− 1) + H } . (A9) ˜ c$_{q}$ = ( ˜ X$_{q}$ ) $^{1}$− q = (˜ ν$_{q}$ ) − $^{1}$( ˜ Z$_{q}$ ) $^{1}$− $^{q}$. (A10) 1 T = 1 ˜ c$_{q}$ ∂S$_{q}$ ∂ ˜ U$_{q}$ , = ( k$_{B}$ ˜ X$_{q}$ ) ∂ ˜ X$_{q}$ ∂ ˜ U$_{q}$ , = q k$_{B}$ ˜ γ, (A11) ˜ γ = β q , (A12) Z$_{q}$ ( α ) = Tr [1 − (1 − q ) α H ] 1 / $^{(1}$− q ) , (B1) Q$_{q}$ ( α ) = 1 ν$_{q}$Z$_{q}$ ( α ) Tr { [1 − (1 − q ) α H ] q/ $^{(1}$− q ) Q } , (B2) y − s = 1 Γ( s ) ∫ ∞ 0 u $^{s}$− $^{1}$e − yu du for s > 0 , (B3) y s = i 2 π Γ( s + 1) ∫ C ( − t ) − $^{s}$− $^{1}$e − yt dt for s > 0 , (B4) 21