Universal Fluctuations in Correlated Systems

Abstract

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D-XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated, equilibrium and non-equilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sand-piles, avalanches and granular media in a self organized critical state. We discuss the relationship with both Gaussian and extremal statistics.PACS numbers: 05.40, 05.65, 47.27, 68.35.Rh Self similarity is an important feature of the natural world. It arises in strongly correlated many body systems when fluctuations over all scales from a microscopic length a to a diverging correlation length ξ lead to the appearence of "anomalous dimension" [1] and fractal properties. However, even in an ideal world the divergence of of ξ must ultimately be cut off by a macroscopic length L, allowing the definition of a range of scales between a and L, over which the anomalous behaviour can occur. Such systems are found, for example, in critical phenomena, in Self-Organized Criticality [2,3] or in turbulent flow problems. By analogy with fluid mechanics we shall call these finite size critical systems "inertial systems" and the range of scales between a and L the "inertial range". One of the anomalous statistical properties of inertial systems is that, whatever their size, they can never be divided into mesoscopic regions that are statistically independent. As a result they do not satisfy the basic criterion of the central limit theorem and one should not necessarily expect global, or spatially averaged quantities to have Gaussian fluctuations about the mean value. In Ref.[4](BHP) it was demonstrated that two of these systems, a model of finite size critical behaviour and a steady state in a closed turbulent flow experiment, share the same non-Gaussian probability distribution function (PDF) for fluctuations of global quantities. Consequently it was proposed that these two systems -so utterly dissimilar in regards to their microscopic details -share the same statistics simply because they are critical. If this is the case, one should then be able to describe turbulence as a finite-size critical phenomenon, with an effective "universality class". As, however, turbulence and the magnetic model are very unlikely to share the same universality class, it was implied that the differences that separate critical phenomena into universality classes represent at most a minor perturbation on the functional form of the PDF. In this paper, to test this proposition, we determine the functional form of the BHP fluctuation spectrum and show that it indeed applies to a large class of inertial systems [5].The magnetic model studied by BHP, the spin wave limit to the two dimensional XY (2D-XY) model, is defined by ...

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“…Moreover, we can address the question of the generality of the observed non-analyticity of the large deviation function in systems in permanent regime where the velocity field is unbounded: from our analysis of the Langevin system, we think that such anomalies could be widely encountered in dissipative far-from-equilibrium systems. An interesting perspective is besides to consider extremely correlated noises, in order to check in this context the ideas exposed in [15,16], where some conjectures are made on the limiting behaviours of the pdf of global variables in highly correlated systems.…”

“…The calculation leading to (16) and (18) assumes a priori λ > 0, but it is useful to study the properties of the analytical continuations of these formulae. First, one remarks that the square root in the definition of η does not induce any breaking of analyticity, for it appears always in quantities such cosh(η γ) or sinh(η γ)/η which are entire functions of λ.…”

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

“…It has been observed that very similar distributions occur in two and three dimensional Ising models at a temperature T * (L) slightly below, but close to the critical temperature [14,15]. We have argued that this is a critical phenomenon [14,16], giving weight to our earlier proposition [12] that many correlated systems, both in and out of equilibrium, can be driven into a state with very similar fluctuations [14,17], irrespectively of their universality class.…”

“…We have argued that this is a critical phenomenon [14,16], giving weight to our earlier proposition [12] that many correlated systems, both in and out of equilibrium, can be driven into a state with very similar fluctuations [14,17], irrespectively of their universality class.…”

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.…”