Nonextensive Statistical Mechanics

84 3 Generalizing What We Learnt0.870.860.850.840.0110.0100.0090.008S qent0.0070.83q ent 0 0.1 0.2 0.3 0.4 0.50.82ω 0ω 00.810.800.790.780 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Fig. 3.21 The ω 0 -dependence of the index q ent in a bosonic d = 2 array of infinite coupled harmonicoscillators at T = 0. Inset: the ω 0 -dependence of the q-entropic density s qent .S qent (L) ∝ L d (d ≥ 1; L →∞) , (3.154)i.e., the thermodynamic extensivity of the entropy is recovered. The index q ent isexpected to depend on some generic parameters (symmetries, gaps, etc), but alsoon the dimension d. In particular, since the exponent (d − 1) in Eq. (3.153) and theexponent d in Eq. (3.154) become closer and closer in the limit d →∞, we expectlim d→∞ q ent (d) = 1. As mentioned before, it is along lines such as this one that atransparent justification could be found for the current use of BG statistical mechanicsin systems like spin-glasses in the mean-field approximation (replica trick).3.4 q-Generalization of the Kullback–Leibler Relative EntropyThe Kullback–Leiber entropy introduced in Section 2.2 can be straightforwardlyq-generalized [88, 92]. The continuous version becomes∫I q (p, p (0) ) ≡−With r > 0wehavethat[ p(0) ] ∫dx p(x)ln q =p(x)dx p(x) [p(x)/p(0) (x)] q−1 − 1.q − 1(3.155)