Magnetic fluctuations in the classicalXYmodel: The origin of an exponential tail in a complex system

Abstract

We study the probability density function for the fluctuations of the magnetic order parameter in the low-temperature phase of the XY model of finite size. In two dimensions, this system is critical over the whole of the low-temperature phase. It is shown analytically and without recourse to the scaling hypothesis that, in this case, the distribution is non-Gaussian and of universal form, independent of both system size and critical exponent eta. An exact expression for the generating function of the distribution is obtained, which is transformed and compared with numerical data from high-resolution molecular dynamics and Monte Carlo simulations. The asymptotes of the distribution are calculated and found to be of exponential and double exponential form. The calculated distribution is fitted to three standard functions: a generalization of Gumbel's first asymptote distribution from the theory of extremal statistics, a generalized log-normal distribution, and a chi(2) distribution. The calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4, despite the fact that critical fluctuations are limited to D=2. These results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state.

Vortex corrections to universal scaling of magnetic fluctuations in 2D XY model

“…In Refs. [5,6] we find analytically that σP (m), shown as the solid line in Fig. 1, is a universal scaling function, not only of system size, but also of temperature and therefore of critical exponent η.…”

“…The former have been discussed in detail in Ref. [5,21]; at fixed temperature, they give small deviations from the universal curve for small system size, L. These are finite size corrections to the thermodynamic limit function and disappear with relatively modest increase in the system size.…”

“…This universal scaling function, plotted as the solid line in Fig. 1, arises when σP (m) is plotted against µ = (m − m )/σ, where m and σ are the mean and standard deviation of the distribution [12,5,13]. It is asymmetric, with an exponential tail for fluctuations below the mean and double exponential for fluctuations above the mean.…”

“…Evidence from, e.g. the Ising model [2,3] and Potts models [4] suggests that this is indeed generally the case.We have, however recently considered an exception to this established phenomenology, the 2D XY model [5,6]. Here the low temperature phase consists of a line of critical points, with one independent critical exponent η(T ) = T /2πJ (with J the exchange constant) extending down to zero temperature and separated from the high temperature paramagnetic phase by the Kosterlitz-Thouless-Berezinskii (KTB) phase transition [7,8,9].…”

Small-World Synchronized Computing Networks for Scalable Parallel Discrete-Event Simulations

“…[36,[54][55][56][57][58][59][60]. For the original PDES scheme (p=0, regular lattice synchronization) exhibiting a KPZ-like rough (or critical) surface, we illustrated (Sect.…”