Criterion for universality-class-independent critical fluctuations: Example of the two-dimensional Ising model

Abstract

Order parameter fluctuations for the two dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T * (L) and of magnetic fields B * (L) are identified, for which the probability density function is similar to that for the 2D-XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality class dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.

Space-time thermodynamics and subsystem observables in a kinetically constrained model of glassy materials

“…Because of the coexisting inactive phase, these distributions display non-Gaussian tails. This behaviour is analogous to the situation in standard phase transitions: the distribution of cavity sizes in a fluid near liquid vapour coexistence has non-Gaussian tails [4], as does the distribution of box magnetization for the Ising model in the vicinity of its phase transition [5].…”

Origin of the approximate universality of distributions in equilibrium correlated systems

“…The main result of ref. [8] is that along the locus of temperatures T * (L), while the correlation length diverges with L as expected for a critical phenomenon, its amplitude is small compared to L: ξ/L ≃ 0.03. Therefore we have R L (T * (L)) ≪ 1 and we expect that the fluctuations at the integral scale l = L could be described, to an excellent approximation, by a perturbative development.…”

“…In this paper we expose these criteria through the study of a well known and well controlled equilibrium system, the 2D Ising model, the aim being to show microscopically how a XY-like behaviour appears in the Ising model. Using numerical simulations for the 2D Ising model [2,8] we established that there exists a range of temperatures T * (L), or applied field H * (L) close to the critical point( 1 ) (T c (L), H = 0), where L is the system size, where the PDF is similar to the BHP function. At this specific temperature or field the magnetization shows intermittent behaviour, where coherent structures appear on intermediate time scales, in analogy with injected power fluctuations in enclosed turbulent flow [3,8].…”

“…Using numerical simulations for the 2D Ising model [2,8] we established that there exists a range of temperatures T * (L), or applied field H * (L) close to the critical point( 1 ) (T c (L), H = 0), where L is the system size, where the PDF is similar to the BHP function. At this specific temperature or field the magnetization shows intermittent behaviour, where coherent structures appear on intermediate time scales, in analogy with injected power fluctuations in enclosed turbulent flow [3,8]. This should be contrasted with the behaviour at the critical point, where the amplitude of the structures and the ensuing intermittency are cut off by the boundaries of the available phase space.…”

“…The second and approximate equality is obtained using the scaling hypothesis and is true independently of ( 1 )T * was defined in ref. [8] as the temperature where the kurtosis of the distribution most closely approximates to that for the BHP function. As usual for finite size systems, we define the critical temperature the one corresponding to the maximum value of the susceptibility.…”