Nonextensive Statistical Mechanics

250 7 Thermodynamical and Nonthermodynamical Applicationsab1.01.0C(n+n w , n w )N=10000a=0.2 ; f=0.01 ; s 1 =1n w =250n w =500n w =1000n w =2000n w =5000C n (n+n w , n w )0.10.110 0 10 2 10 3 10 4 10 5 10 –3 10 –2 10 –1 10 0 10 1 10 2ncdN=10000a=0.2 ; f=0.01 ; s 1 =1n w =250n w =500n w =1000n w =2000n w =5000n/n w1.05C(n+n w , n w )0.110 110 1 10 41.01.032 x 32 latticek=132 x 32 latticen w =250n w =500n w =1000n w =2000n w =5000C n (n+n w , n w )n w =250n w =500n w =1000n w =2000n w =500010 2n10 30.110 –3 10 –2 10 –1 10 0n/n1.05w10 110 2Fig. 7.38 The collapsed and noncollapsed correlation functions for the Newman model (top) andthe Olami–Feder–Christensen model (bottom) (from [381]).Let us address now the most classical quantity for earthquakes, namely the probabilityof having earthquakes of magnitude m (Gutenberg–Richter law). A nontrivialresult (generalizing in fact the classical Gutenberg–Richter law) has been analyticallyobtained [542] along this line for the cumulative probability G(> m) involvingtwo parameters, q and a (a is the constant of proportionality between the releasedrelative energy ɛ and the linear dimension r of the fragments of the fault plates).These results are much in line with those presented in Figs. 7.39 and 7.40.Finally, let us focus on the histograms of the avalanche size differences (returns,as such quantities are called in finance). These have been focused in [855,856]. In particular, such probability distributions have been calculated in a dissipativeOlami–Feder–Christensen model (Fig. 7.41), and also for real earthquakes(Fig. 7.42). The results for the OFC model have been calculated in both a