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 ...

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 LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.

Bramwell et al. Reply: Zheng and Trimper [1] confirm the conjecture given in our Letter [2] that the probability distribution for order parameter fluctuations in the 2D and 3D Ising models at a temperature T ‫ء‬ ͑L͒ slightly below T C ͑L !`͒ approximates the universal functional form of the 2D-XY model in its low temperature phase [3]. They show quantitatively that T C 2 T ‫ء‬ ͑L͒ scales as L 21͞n . The XY -type scaling is, of course, only one locus in the L 21 , T plane and for general L 21 , T the probability distribution function (PDF) is not of the XY form [2,4]. The point of departure between our interpretation of this result and that of Zheng and Trimper is that we attribute the PDF to critical fluctuations and they do not. We are pleased to take the opportunity to discuss this point in detail.The 2D-XY model is critical throughout the low temperature regime with diverging longitudinal fluctuations. It is therefore incorrect to think that critical "fluctuations are mainly rotational." The physics of a phase transition in a spin system with continuous symmetry is controlled by the divergence of the longitudinal susceptibility, the transverse susceptibility being infinite at all temperatures. The lengthening magnetization vector as order develops drives the diffusion constant around the circle (in the XY model) to zero in the thermodynamic limit, and consequently rotational symmetry is broken. The scalar magnetization is therefore a critical quantity, as can be seen through any finite-size scaling criterion. However, it is rather a special limit for critical fluctuations: despite the susceptibility diverging as ϳL d2h and s͗͞m͘ remaining independent of system size, the latter ratio is small, ഠh͞4 [3]. The result, paradoxically, is that the divergent fluctuations never bring the order parameter near the limits m 0 and m 1. The critical fluctuations therefore occur without ever changing the fixed topology (or symmetry) imposed by the corrections to the thermodynamic limit [3]. That is, there is a barrier to jump to arrive at m 0, but ͗m͘ ϳ L 2h͞2 is not an intensive variable and the free energy barrier is not extensive; it is a correction to the thermodynamic limit and a pure effect of criticality.Zheng and Trimper find that, for the Ising model, the measured correlation length at T ‫ء‬ ͑L͒ is small compared with L. This property should ensure that fluctuations in the Ising systems studied can be described in a similar way to those of the XY model. Indeed the authors point out that at T ‫ء‬ ͑L͒, the order parameter remains far from the minimum of probability: m 0. However, they are wrong to conclude that the fluctuations at this temperature are "not a characteristic property at the critical point": the observation of the universal fluctuations at constant s Lt means that the "small" correlation length is fixed by the system size; it does diverge in the thermodynamic