Nonextensive Statistical Mechanics

196 5 Deterministic Dynamical Foundations of Nonextensive Statistical MechanicsPDFPDFTemperature10 010 –110 –210 –310 010 –110 –210 –30,500,480,460,440,420,400,38class 1 event–6 –4 –2 0 2 4 6(y-)/σclass 3 event–6 –4 –2 0 2 4 6(y-)/σn = 200 n = 1000 n = 2000δ = 100 δ = 100 δ = 10010 –110 –1PDF10 010 –210 –3–6 –4 –2 0 2 4 6(y-)/σPDF10 010 –210 –310 –110 –210 –3–6 –4 –2 0 2 4 6(y-)/σn = 200 10 0n = 1000 10 0n = 2000δ = 100δ = 100δ = 100PDF10 –110 –210 –3–6 –4 –2 0 2 4 6(y-)/σ–6 –4 –2 0 2 4 6(y-)/σ10 0 10 1 10 2 time 10 3 10 4 10 5PDFQSS regimeclass 3 eventclass 1 eventequilibriumFig. 5.45 We show the time evolution of two events with the same M 1 initial conditions, moreprecisely with N=20,000 at U = 0.69, but belonging to different classes (1 and 3); the durationalong which we are doing the CLT sums is n × δ. The evolution towards the final attractor appearsto be a Gaussian for the event of class 3 and a q-Gaussian-like for the event of class 1. The latter isgiven by G q (x) = A(1 − (1 − q)βx 2 ) 1/1−q , with q = 1.42 ± 0.1, β = 1.3 ± 0.1, and A = 0.55.Notice that the tails emerge clearly while increasing the number n of summands. This is also truein the case of the Gaussian, as predicted by the CLT. From [48] (see also [47]).5.7 A Conjecture on the Time and Size Dependences of EntropyWe have seen that, for a (not yet fully qualified) large class of systems, there isa special value of q, q N , such that S qN (N, t) ∝ N (N →∞). This is so for allvalues of time t, including t →∞, if we are describing the system within somefinite resolution (or some finite degree of fine-graining, i.e., ɛ>0). We have alsoseen that, for a (once again not yet fully qualified) large class of systems, there isa special value of q, q t , such that S qt (N, t) ∝ t (t →∞). This is so for an infiniteresolution (or ideally precise degree of fine-graining, i.e., ɛ = 0). The scenario isschematically indicated in Fig. 5.64. If this scenario is correct, then we conjecturethat q N = q t ≡ q ent , hence, in the ɛ → 0 limit, we would generically have thefollowing form:S qent ∼ sNt (N →∞; t →∞; s ≥ 0) . (5.59)