Nonextensive Statistical Mechanics

188 5 Deterministic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 5.37 Time evolution ofthe velocity probabilitydistribution function (PDF)for u(= U) = 0.69 anddifferent sizes. (a) Attimet = 0 we start with simpleWBIC, or DWBIC, velocityPDF. (b) In the transientregime, where T shows aplateau corresponding toT QSS and the system lives in aQSS, the velocity PDFs donot change in time and arevery different from theGaussian BG canonicalequilibrium distribution (fullcurve). The PDFs at t = 1200and N = 1000, 10, 000, and100, 000 show a convergencetowards a non-Gaussian PDF.(c) We show the numericalPDFs at t = 500, 000 forN = 500 and 1000. We getan excellent agreement withthe Maxwellian BG canonicalequilibrium distribution atT = 0.476 (from [373]).in [178], and are illustrated in Figs. 5.49, 5.50, and 5.51. They are completely analogousto that of the d = 1 case, and strongly suggest that the relevant exponent κdoes not depend separately on α and d, but, like N ⋆ (see Eq. (3.69)), only on theratio α/d.The above molecular dynamical results concerned the disordered (paramagnetic)phase. Also are available results [375, 376] for the ordered (ferromagnetic) phaseof the d = 1 model, more precisely for its QSS. For reasons that are not totallytransparent, the value for κ obtained on the QSS (below u c ), turns out to numericallybe 1/3 of its value above u c : See Fig. 5.52.5.4.3 Aging and Anomalous DiffusionThe very fact that, for u < u c and fixed N, a QSS exists which, after a time ofthe order of τ QSS , eventually goes to thermal equilibrium implies that the systemhas some sort of internal clock. This immediately suggests that aging should beexpected. More precisely, if we consider a two-time autocorrelation function C(t +t W , t W ) of some dynamical variables of the system, we expect this quantity to dependnot only on time t, but also on the waiting time t W . This is precisely what is verified