Cumulant Ratios Research Articles

Statistical and dynamical fluctuations in the ratios of higher net-proton cumulants in relativistic heavy-ion collisions

With the help of transport and statistical models, we find that the ratios of higher net-proton cumulants measured at RHIC are dominated by Poisson-like statistical fluctuations. A way to eliminate this statistical fluctuation is suggested. The obtained dynamical ratios of higher net-proton cumulants are demonstrated to be more relevant to the underlying physics, i.e. the correlations between proton and antiproton, or the critical fluctuations.

A Unique Property of the Normal Distribution Associated With Perturbing a General Random Variable

This article shows that when an independent perturbation is added to any nonnormal variable with finite moments the resulting variable is closer to normal when measured by each nonzero cumulant ratio. We also show that the normal density function is the only form of density with this property. This result provides weak evidence that the mixing of subpopulations of subjects can promote normality in the overall population.

Possibility of Fisher renormalization of the critical exponents in an Ising fluid

Using Monte Carlo simulation techniques, we study the ferromagnetic order-disorder phase transition in Ising spin fluids with hard-core Yukawa interaction truncated at various cutoff radii r{c}. We focus our interest on the dependence of critical quantities such as the Binder cumulant and various exponent ratios on the value of r{c}, and on the question whether the Fisher-renormalized exponents expected for such systems can be observed in the simulations. It turns out that the corrections to scaling decaying with a rather small exponent prevent reaching the asymptotic region with the computational power available. Thus, we observe only effective exponents, with different (nonuniversal) values depending on the cutoff radius. The same behavior is also found for the critical Binder cumulant. Nevertheless, an exact investigation of the effective susceptibility exponent gamma{eff} as a function of temperature seems to point towards a Fisher-renormalized value. For two selected cutoff radii, the critical temperature is determined more accurately using, in addition to the cumulant crossing technique, the scanning technique and the shifting technique, taking into account corrections to scaling. Simulations of Ising fluids with constant cutoff radius and varying Yukawa-tail screening lengths lambda also show a nonuniversal dependence of U{c} on lambda. Finally, we have performed simulations of the Ising lattice model with increasing number of couplings which show the expected asymptotic behavior, independent of the range of interactions.

NONEQUILIBRIUM MULTICRITICAL BEHAVIOR IN ANISOTROPIC HEISENBERG FERROMAGNET DRIVEN BY OSCILLATING MAGNETIC FIELD

The Heisenberg ferromagnet (uniaxially anisotropic along z-direction), in the presence of time dependent (but uniform over space) magnetic field, is studied by Monte Carlo simulation. The time dependent magnetic field was taken as elliptically polarised in such a way that the resulting field vector rotates in the XZ-plane. In the limit of low anisotropy, the dynamical responses of the system are studied as functions of temperature and the amplitudes of the magnetic field. As the temperature decreases, it was found that the system undergoes multiple dynamical phase transitions. In this limit, the multiple transitions were studied in details and the phase diagram for this observed multicritical behavior was drawn in field amplitude and temperature plane. The natures (continuous/discontinuous) of the transitions are determined by the temperature variations of fourth-order Binder cumulant ratio and the distributions of order parameters near the transition points. The transitions are supported by finite size study. The temperature variations of the variances of dynamic order parameter components (for different system sizes) indicate the existence of diverging length scale near the dynamic transition points. The frequency dependences of the transition temperatures of the multiple dynamic transition are also studied briefly.

Nonequilibrium dynamic phase transition in a kinetic Ising spin system

We studied within the framework of a mean-field approach the nonequilibrium dyn amic phase transition of a kinetic Ising spin system subject to a perturbative f ield and temperature simultaneously by comparison between time-dependent Ginzbur g-Landau and Glauber dynamics models. The dynamic phase transition (DPT) boundar ies, separating a symmetry-breaking dynamic ordered phase from its symmetric dyn amic disordered counterpart, were identified through a systematic simulation of the above two models. The dependence of the dynamic order parameter Q and the fo urth order cumulant ratio U upon the temperature t(r0), the frequency ω and amplitude h0 of driving field were also investigated in detai l. A discussion was presented concerning the current controversies on whether bo th a discontinuous dynamic phase transition occurs possibly below a specific low temperature and a tri-critical point exists on the DPT boundary in a kinetic Is ing spin system.

Hadronic fluctuations at the QCD phase transition

We discuss the properties of hadronic fluctuations, i.e., fluctuations of net quark and isospin numbers as well as electric charge, in the vicinity of the QCD transition in isospin-symmetric matter at vanishing quark chemical potential. We analyse second- and fourth-order cumulants of these fluctuations and argue that the ratio of quartic and quadratic cumulants reflects the relevant degrees of freedom that carry the quark number, isospin or charge, respectively. In the hadronic phase we find that an enhancement of charge fluctuations arises from contributions of doubly charged hadrons to the thermodynamics. The rapid suppression of fluctuations seen in the high-temperature phase suggests that in the QGP, net quark number and electric charge are predominantly carried by quasi-particles, with the quantum numbers of quarks.

Numerical method for accessing the universal scaling function for a multiparticle discrete time asymmetric exclusion process

In the universality class of the one-dimensional Kardar-Parisi-Zhang (KPZ) surface growth, Derrida and Lebowitz conjectured the universality of not only the scaling exponents, but of an entire scaling function. Since and Derrida and Lebowitz's original publication [Phys. Rev. Lett. 80, 209 (1998)] this universality has been verified for a variety of continuous-time, periodic-boundary systems in the KPZ universality class. Here, we present a numerical method for directly examining the entire particle flux of the asymmetric exclusion process (ASEP), thus providing an alternative to more difficult cumulant ratios studies. Using this method, we find that the Derrida-Lebowitz scaling function (DLSF) properly characterizes the large-system-size limit (N--> infinity) of a single-particle discrete time system, even in the case of very small system sizes (N< or =22). This fact allows us to not only verify that the DLSF properly characterizes multiple-particle discrete-time asymmetric exclusion processes, but also provides a way to numerically solve for quantities of interest, such as the particle hopping flux. This method can thus serve to further increase the ease and accessibility of studies involving even more challenging dynamics, such as the open-boundary ASEP.

On the high order multiplicity moments

The description of multiplicity distributions in terms of the ratios of cumulants to factorial moments is analysed both for data and for Monte Carlo generated events. For the PYTHIA generated events, the moments are investigated for the restricted range of phase space and for the jets reconstructed from single particle momenta. The results cast doubts on the validity of extended local parton–hadron duality and suggest the possibility of more effective experimental investigations concerning the origin of the observed structure in the dependence of moments on their order.

Nonuniversal finite-size scaling in anisotropic systems

We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.

Time series nonlinearity modeling: A Giannakis formula type approach

In this paper, we propose a method for identifying the coefficients of a simplified Second Order Volterra Model (SOVM) driven by a normal i.i.d. white noise. The interest of estimating the coefficients of such a model is to easily model nonlinear time series by identifying a linear spectrum and a nonlinear spectrum. In fact, the nonlinear spectrum is the spectrum of output data of a quadratic system (squarer) driven by a normal i.i.d. white noise while the linear spectrum is the output data spectrum of a linear system driven by the same noise. Consequently, by estimating the linear and nonlinear spectrum components, the proposed algorithm locates (in the Fourier domain) and quantifies the nonlinear artifacts in an observed time series, this observed time series being the output of a nonlinear system and the input data of this system not being available. The method for estimating the model coefficients is quite simple and is based on the ratio of products of Higher Order Cumulants. For this reason, the method of identification is close to Giannakis’ formula which identifies the coefficients of a linear system driven by a non symmetric noise and also uses the ratio of cumulants. In this paper, we also address the question of order selection of both parts of the simplified SOVM (i.e. the linear and quadratic parts) based on hypothesis testing, the order of each part interfering strongly in the final results. Finally, we propose a method for verifying that the higher order statistics (HOS) of the observed time series are matched with the HOS derived from the estimated coefficients, thus proving that the time series is well modeled by the estimated nonlinear parametric model.

Absence of first-order transition and tricritical point in the dynamic phase diagram of a spatially extended bistable system in an oscillating field.

It has been well established that spatially extended, bistable systems that are driven by an oscillating field exhibit a nonequilibrium dynamic phase transition (DPT). The DPT occurs when the field frequency is of the order of the inverse of an intrinsic lifetime associated with the transitions between the two stable states in a static field of the same magnitude as the amplitude of the oscillating field. The DPT is continuous and belongs to the same universality class as the equilibrium phase transition of the Ising model in zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed that the DPT becomes discontinuous at temperatures below a tricritical point [M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on observations in dynamic Monte Carlo simulations of a multipeaked probability density for the dynamic order parameter and negative values of the fourth-order cumulant ratio. Both phenomena can be characteristic of discontinuous phase transitions. Here we use classical nucleation theory for the decay of metastable phases, together with data from large-scale dynamic Monte Carlo simulations of a two-dimensional kinetic Ising ferromagnet, to show that these observations in this case are merely finite-size effects. For sufficiently small systems and low temperatures, the continuous DPT is replaced, not by a discontinuous phase transition, but by a crossover to stochastic resonance. In the infinite-system limit, the stochastic-resonance regime vanishes, and the continuous DPT should persist for all nonzero temperatures.

Tests of parameters of several gamma distributions with inequality restrictions

Gamma distribution is one of the most used methods of modeling lifetime data. However, testing homogeneity of parameters of m ≥ 3 gamma distributions against order restrictions is almost non-existent in the current literature. We propose two methods to this end: one uses quadratic forms involving ratios of cumulants as test statistic and the other is a stepwise procedure which uses Fisher's method of combining p-values when shape parameters are equal but unknown. Both procedures allow use of arbitrary sample sizes of m populations. Test of the inequality restrictions as a null hypothesis against unrestricted alternatives is also considered. A Monte Carlo study of power at various alternatives shows that both methods are competitive when they are applicable.

Cumulant ratios in fully developed turbulence

In the context of random multiplicative cascade processes, we derive analytical solutions for one- and two-point cumulants with restored translational invariance. On taking ratios of cumulants in ln ε, geometrical effects due to spatial averaging cancel out. These ratios can successfully distinguish between splitting functions while multifractal scaling exponents and multiplier distributions cannot.

Cumulant ratios and their scaling functions for ising systems in a strip geometry

We calculate the fourth-order cumulant ratio (proposed by Binder) for the two-dimensional Ising model in the strip geometry Lxinfinity. The density-matrix renormalization-group method enables us to consider typical open boundary conditions up to L=200. Universal scaling functions of the cumulant ratio are determined for strips with parallel as well as opposing surface fields. Their asymptotic properties are also examined.

DYNAMICS OF EDEN CLUSTERS: CONFIRMATION OF APPERT–DERRIDA UNIVERSALITY

Universality of ratios of cumulants depends on the minor changes in the definition of time.

Large deviation function for the eden model and universality within the one-dimensional kardar-parisi-zhang class

It has been recently conjectured that for large systems, the shape of the central part of the large deviation function of the growth velocity would be universal for all the growth systems described by the Kardar-Parisi-Zhang equation in 1+1 dimension. One signature of this universality would be that the ratio of cumulants R(t)=[<h(3)(t)>(c)](2)/[<h(2)(t)>(c)<h(4)(t)>(c)] would tend towards a universal value 0.415 17ellipsis as t tends to infinity, provided periodic boundary conditions are used. This has recently been questioned by Stauffer. In this paper we summarize various numerical and analytical results supporting this conjecture, and report in particular some numerical measurements of the ratio R(t) for the Eden model.

Scaling and finite-size scaling in the two-dimensional randomly coupled Ising ferromagnet

It is shown by Monte Carlo method that the finite size scaling (FSS) holds in the two dimensional random-coupled Ising ferromagnet. It is also demonstrated that the form of universal FSS function constructed via novel FSS scheme depends on the strength of the random coupling for strongly disordered cases. Monte Carlo measurements of thermodynamic (infinite volume limit) data of the correlation length ($\xi$) up to $\xi \simeq 200$ along with measurements of the fourth order cumulant ratio (Binder's ratio) at criticality are reported and analyzed in view of two competing scenarios. It is demonstrated that the data are almost exclusively consistent with the scenario of weak universality.

Universality of the growth velocity distribution in 1+1 dimensional growth models

In order to gain some insight into the scaling properties of the Kardar–Parisi–Zhang (KPZ) equation, the large deviation function f(v) for the distribution of growth velocities was calculated for one particular model of the KPZ class, the asymmetric exclusion process (ASEP) [1,2]. First we shall summarize some analytical results. For large systems, the shape of f(v) is independent of the microscopic details of the model within a scaling region. We conjecture that it is universal to all models in the KPZ class. This would imply that some special ratios of velocity cumulants are also universal. In order to check the theory against numerical measurements, some Monte Carlo simulations have been performed for the ASEP with different random-number generators.

Large deviation function of the partially asymmetric exclusion process.

The large deviation function obtained recently by Derrida and Lebowitz [Phys. Rev. Lett. 80, 209 (1998)] for the totally asymmetric exclusion process is generalized to the partially asymmetric case in the scaling limit. The asymmetry parameter rescales the scaling variable in a simple way. The finite-size corrections to the universal scaling function and the universal cumulant ratio are also obtained to the leading order.

Cumulant Ratios of Eden Model Surfaces in 1+1 Dimensions

Monte Carlo simulations suggest that dimensionless ratios of cumulants for the height of the surface depend on how randomness is used.