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 ...

Can one construct a thermodynamics for compact, slowly moving powders and grains? A few years ago, Edwards proposed a possible step in this direction, raising the fascinating perspective that such systems have a statistical mechanics of their own, different from that of Maxwell, Boltzmann, and Gibbs, allowing us to have some information while still ignoring dynamic details. Recent developments in the theory of glasses have come to confirm these ideas within mean field. In order to go beyond, we explicitly generate Edwards' measure in a 3D model. Comparison of the results with the irreversible compaction data shows very good agreement. The present framework immediately suggests new experimental checks.

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.

The nonequilibrium critical dynamics of the 2D XY model is investigated numerically through Monte Carlo simulations and analytically in the spin-wave approximation. We focus in particular on the behaviour of the two-time response and correlation functions and show that the ageing dynamics depends on the initial conditions. The presence of critical fluctuations leads to non-trivial violations of the fluctuation-dissipation theorem apparently reminiscent of the three dimensional EdwardsAnderson spin glass model. We compute for this reason the finite-size overlap probability distribution function and find that it is related to the finite-time fluctuation-dissipation ratio obtained in the out of equilibrium dynamics, provided that the temperature is not very low. Il y a du miracle dans toute coïncidence, M.Yourcenar.