Nonextensive Statistical Mechanics

7.4 Computer Sciences 273Fig. 7.71 Cumulative distribution of the scaled gross domestic product of 167 countries aroundthe world for the year 2000; q = 3.5. See [596] for further details.and q A < 1. A very convenient random number generator following a q-Gaussiandistribution has been recently proposed in [627].Part of the simulated annealing procedure consists in the Cooling algorithm,which determines how the effective temperature T is decreased with time, so thatthe global minimum is eventually attained within the desired precision. A quickcooling is of course computationally desirable. But not too quick, otherwise the rateof success of ultimately arriving to the real global minimum decreases sensibly. Theoptimal cooling procedure appears to be given by [625, 626]T (t)T (1) = 2q V −1 − 1(1 + t) q V −1− 1 = ln q V(1/2)ln qV [1/(t + 1)](t = 1, 2, 3,...) , (7.26)where T (1) is the initial high temperature imposed onto the system. We verify that,for q V = 1, we haveT (t)T (1) = ln 2ln(1 + t)(t = 1, 2, 3,...) , (7.27)and that, for q V = 2, we haveT (t)T (1) = 2 t(t = 1, 2, 3,...) . (7.28)For the D = 1 upper limit, we have q V = 3, henceT (t)T (1) = 3(1 + t) 2 − 1(t = 1, 2, 3,...) . (7.29)