Nonextensive Statistical Mechanics

5.7 A Conjecture on the Time and Size Dependences of Entropy 19710 010 –1Time averageN = 10000 - U = 0.69 - QSSn = 1000δ = 100q-Gaussian q = 1.42 β = 1.3Gaussian(a)10 010 –1Time averageN = 20000 - U = 0.69 - QSSn = 2000δ = 100q-Gaussian q = 1.42 β = 1.3Gaussian(b)PDF10 –2PDF10 –210 –310 –310 –410 –410 –5–10 –8 –6 –4 –2 0 2 4 6 8 10(y - )/σ10 –5–10 –8 –6 –4 –2 0 2 4 6 8 10(y - )/σ10 010 –1Time averageN = 50000 - U = 0.69 - QSSn = 5000δ = 100q-Gaussian q = 1.42 β = 1.3Gaussian(c)10 010 –1Time averageN = 100000 - U = 0.69 - QSSn = 10000δ = 100q-Gaussian q = 1.42 β = 1.3Gaussian(d)PDF10 –2PDF10 –210 –310 –310 –410 –410 –5–10 –8 –6 –4 –2 0 2 4 6 8 10(y - )/σ10 –5–10 –8 –6 –4 –2 0 2 4 6 8 10(y - )/σFig. 5.46 For U = 0.69 we show the PDFs obtained considering single events of class 1 for theHMF system sizes N = 10, 000, 20, 000, 50, 000 and 100, 000, with δ = 100 and n = N/10.Again the indications for a q-Gaussian-like attractor becomes stronger and stronger when sendingboth N and n to infinity. Notice that we consider here a larger scale (compared to that of Fig. 5.45)in the ordinate in order to see in detail the tails of the PDF. The same q-Gaussian reported in theprevious figure, with A=0.55, q = 1.42 ± 0.1, and β = 1.3 ± 0.1 and obtained by fitting thecase with N = 100, 000, is here shown for comparison, together with the standard Gaussian withunitary variance. From [48] (see also [47]).Fig. 5.47 The u ≡ E N /NN ⋆ - dependence of the properly scaled maximal Lyapunov exponent˜λ maxN,forthed = 1 α − XY model and typical values of N, forα = 1.5 (a) andα = 0.2(b). As illustrated in Fig. 5.48, the N →∞limit yields, for high enough values of u (in factfor u > u c = 0.75 , ∀α), a nonvanishing (vanishing) value for ˜λ maxNfor α ≥ 1(0≤ α ≤ 1)(from [177]).