Nonextensive Statistical Mechanics

204 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsγ2/(3–q)1,71,61,51,41,31,21,11,00,90,80,70,6α-XY modelN = 5000U = 0.69 N = 2000N = 1000α = 0α = 00,51,0 1,1 1,2 1,3 1,4 1,5 1,6Anomalous diffusion exponentαM 0= 0but different i.c.inside [0,1],M 0= 1M M 0 M 0= 0 = 0.2 0= 0.4M 0= 1α = 1 α = 0.8α = 0.5M α = 00= 0.8α = 0.2γFig. 5.61 For different system sizes and initial conditions, and for several values of the parameterα which fixes the range of the interaction of a generalized version of the HMF model [12], thefigure illustrates the ratio of the anomalous diffusion exponent γ divided by 2/(3 − q) vs.γ .Theentropic index q is extracted from the relaxation of the correlation function (see previous figure).This ratio is always one within the errors of the calculations (from [41]).10,8Equilibrium regime(a)1U=0.69QSS regime(b)0,6N=10000,40,2polarization pmagnetization M0,1polarization pmagnetization MN –1/600 0,2 0,4 0,6 0,8 1U10 3 10 4 10 5 10 6NFig. 5.62 (a) The magnetization M and the polarization p are plotted vs. the energy density forN = 10, 000 at equilibrium: the two-order parameters are identical. (b) The same quantities plottedin (a) are here reported vs. the size of the system, but in the metastable QSS regime. In this case,increasing the size of the system, the polarization remains constant around a value p ∼ 0.24 whilethe magnetization M goes to zero as N −1/6 (from [41]).