Nonextensive Statistical Mechanics

5.6 Connection with Critical Phenomena 195HMF ModelN = 20,000 - U = 0.69Single Eventsn = 2000 δ = 100trans. = 200PDF10 010 –1y_i = 1/sqrt(n) Sum_k p_i(k)i = 1,...,N k = 1,...,nM1ic - ev.of class 1q-Gaussian (q = 1.5)GaussianM0ic10 –210 –3Temperature0,500,480,460,440,420,400,38–6 –4 –2 0 2 4 6y/σEquilibriumtotal y-Sum calculation time = 200000QSSM1ic - event of class 1M0ic - typical event10 0 10 1 10 2 10 3 10 4 10 5timeFig. 5.44 (a) Comparison of the Central Limit Theorem behavior for the u = 0.69, N = 20, 000case with initial magnetization m = 1andm = 0. A Gaussian (dashed curve) with unit varianceand a q-Gaussian with A = 0.66, q = 1.5, and β = 1.8 (full curve) are also reported forcomparison. (b) Temperature time evolutions of the same events shown in panel (a) (from [46]).5.6 Connection with Critical PhenomenaSince it is since long known that systems at criticality (in the sense of standardsecond order critical phenomena) exhibit a fractal geometry, it is kind of naturalto expect that connections would exist between q and the critical phenomenon:see [353–355]. In particular, an interesting analytical connection has already beenestablished for the Ising ferromagnet, namely [354]q = 1 + δ2, (5.58)where δ is the critical exponent characterizing the dependance, at precisely the criticalpoint, of the order parameter with its thermodynamically conjugate field (e.g.,M ∼ H 1/δ , where M and H are, respectively, the magnetization and the externalmagnetic field).