Nonextensive Statistical Mechanics

246 7 Thermodynamical and Nonthermodynamical ApplicationsFig. 7.31 Dependence of (q,τ 0 )onm th . Data from the California catalog. From the bottom to thetop, m th = 0.0, 1.4, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5. For further details see [544].Moreover, the pair (q,τ 0 ) depends in a defined form on the threshold m th ,asindicated in Fig. 7.31.As another important application to earthquakes let us focus on the aging whichoccurs within the nonstationary regime called Omori regime, the set of aftershocksthat follow a big event. We introduce the following correlation function:whereC(n + n W , n W ) ≡ 〈t n+n Wt nW 〉−〈t n+nW 〉〈t nW 〉σ n+nW σ nW, (7.15)〈t m 〉= 1 N∑N−1t m+k , (7.16)k=0〈t m t m ′〉= 1 N∑N−1t m+k t m ′ +k , (7.17)k=0andσ 2 m =〈t 2 m 〉−〈t m〉 2 , (7.18)N being the number of events that are being considered within the Omori regime,and t m being the time at which the mth event occurs; m is sometimes referred to asnatural time. By definition, C(n W , n W ) = 1. For a stationary state, C(n + n W , n W )depends on the natural time n, but not on the waiting natural time n W ;ifitalsodepends on n W , the state is necessarily a nonstationary one, and exhibits aging,one of the most characteristic features of glassy systems. The correlation functionof typical earthquakes in Southern California has been discussed in [541]:see Figs. 7.32 and 7.33 for catalog data, respectively, inside and outside the Omori