Nonextensive Statistical Mechanics

150 4 Stochastic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 4.28 A typical vortex configuration in a 256 × 256 n = d = 2 system. The arrow on eachsite represents the order parameter at that point. Not all the lattice sites are shown. The squares andtriangles are in the core regions of +1 and−1 vortices, respectively, where the magnitude of theorder parameter is near zero (from [351]).O(n) symmetry (n = 2 corresponds to the XY model, n = 3 corresponds to theHeisenberg model, and so on; the analytic limit n → 1 would yield the Ising model).We specifically address the kinetics of point defects (see the vortices in Fig. 4.28)during a quenching from high temperature to zero temperature for the d = n model.The theoretical description is done in terms of a time-dependent Ginzburg–Landauequation (similar to a Langevin equation). As a main outcome, one obtains that thedistribution of the vortex velocity v is, although not written in this manner by theauthors [350, 351], given bywithP(v) ∝ e −|v|2 /v 2 0q , (4.110)q = d + 4d + 2 , (4.111)v 0 being a reference velocity which approaches zero for time increasing after themoment at which the quenching was done. It is certainly very interesting, althoughyet unexplained, to notice that the value of q precisely is the one which separatesthe finite from the infinite variance regions of q at d dimensions (see Eq. (4.30)).