Nonextensive Statistical Mechanics

190 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics10 0ENSEMBLE AVERAGEN = 50,000 - U = 0.69 QSSn = 1t = 200t = 250000(a)PDF10 –110 –210 –3–5 –4 –3 –2 –1 0 1 2 3 4 5(y-)/σ10 0TIME AVERAGEN = 50,000 - U = 0.69 - QSS0.6(b)PDF10 –110 –2n = 5000δ = 100q-Gaussian (q = 1.4)PDF0.40.20.0–2 –1 0 1 2(y-)/σ10 –310 –4–10 –8 –6 –4 –2 0 2 4 6 8 10(y-)/σFig. 5.39 Numerical simulations for the HMF model for N = 50, 000, U = 0.69 and M 1 initialconditions in the QSS regime. (a) We plot the PDFs of single rotor velocities at the times t = 200and t = 250, 000 (ensemble average over 100 realizations). (b) We plot the time average PDF forthe variable y calculated over only one single realization in the QSS regime and after a transienttime of 200 units. In this case, we used δ = 100 and n = 5000, in order to cover a very largeportion of the QSS. Again, a q-Gaussian reproduces very well the calculated PDF both in the tailsand in the central part (see inset). See text for further details (from [45]).5.4.4 Connection with Glassy SystemsWe have seen in the previous subsection that there is aging at the QSS below thecritical point, whereas no such phenomenon survives above u c . We expect then tohave some sort of glassy behavior during the QSS, and no such behavior above u c .This is precisely what we see in Fig. 5.62 (see also [42, 43]).