Nonextensive Statistical Mechanics

5.4 Many-Body Long-Range-Interacting Hamiltonian Systems 189Fig. 5.38 Time evolution of the HMF temperature for the energy density u(= U) = 0.69, N =1000 and several initial conditions with different magnetizations. After a very quick cooling, thesystem remains trapped into metastable long-living Quasi-Stationary States (QSS) at a temperaturesmaller than the equilibrium one. Then, after a lifetime that diverges with the size, the noise inducedby the finite number of spins drives the system towards a complete relaxation to the equilibriumvalue. Although from a macroscopic point of view the various metastable states seem similar, theyactually have different microscopic features and correlations which depend in a sensitive way onthe initial magnetization (from [41]).in [41, 44, 379, 380]: See Figs. 5.53, 5.54 and 5.60. It is quite remarkable that q-exponential decays are observed in these (and other) cases, and that data collapse,in the formC(t + t W , t W ) = e −Bt/t β Wq (B > 0; β ≥ 0) (5.48)is possible (such as in usual spin-glasses). The value q ≃ 2.35 (corresponding tothe (p,θ)-space [44]) is essentially what elsewhere (namely, in the context of theq-triplet to be soon discussed) is noted q rel . Another remarkable fact (see Fig. 5.55)is that, for u > u c , Eq. (5.48) is still satisfied with the same value of q ≃ 2.35, butwith β = 0, i.e., without aging. Let us stress that, for a standard BG system (e.g.,if α/d > 1), one normally observes, both above and below the critical point, q = 1and β = 0.Let us now focus on the diffusion of the angles {θ i } by allowing them to freelymove within −∞ to +∞. The probability distributions, and corresponding anomalousdiffusion exponent γ , can be seen in Figs. 5.56, 5.57, 5.58, 5.59, and 5.60. Fromthe data in Fig. 5.60 we can verify (see Fig. 5.61) the agreement, within a 10% error,with the scaling predicted in Eq. (4.16).For phenomena occurring at the edge of chaos of simple maps and related tothose described above, see [42, 43].