Critical phenomena in a disc-percolation model and their application to relativistic heavy ion collisions

By studying the critical phenomena in continuum-percolation of discs, we find a new approach to locate the critical point, i.e.using the inflection point of P_∞ as an evaluation of the percolation threshold.The susceptibility, defined as the derivative of P_∞, possesses a finite-size scaling property, where the scaling exponent is the reciprocal of ν, the critical exponent of the correlation length.A possible application of this approach to the study of the critical phenomena in relativistic heavy ion collisions is discussed.The critical point for deconfinement can be extracted by the inflection point of P_(QGP)-the probability for the event with QGP formation.The finite-size scaling of its derivative can give the critical exponent ν, which is a rare case that can provide an experimental measure of a critical exponent in heavy ion collisions.     

The effects of surfaces on dynamic critical behavior

The field theoretic renormalization-group approach for the study of critical behavior near free surfaces is generalized to dynamic properties. Time-dependent Ginzburg-Landau models with nonconserved or conserved order parameters — semi-infinite generalizations of the so-called modelsA andB — are considered. The asymptotic behavior of response and correlation functions is analyzed at the ordinary and special transitions in 4-? dimensions, and dynamic scaling laws for surface quantities are obtained. It is shown that the critical exponents can be expressed entirely in terms of static bulk and surface exponents andz, the dynamic bulk exponent. The critical exponents for the leading frequency, temperature and momentum singularities of the surface two-spin correlation function at the ordinary transition differ appreciably from the corresponding bulk analogues. In addition, the shape function which describes its frequency dependence differs qualitatively from the one of the bulk correlation function.     

Critical Dynamics in Thin Films

Critical dynamics in film geometry is analyzed within the field-theoretical approach. In particular we consider the case of purely relaxational dynamics (Model A) and Dirichlet boundary conditions, corresponding to the so-called ordinary surface universality class on both confining boundaries. The general scaling properties for the linear response and correlation functions and for dynamic Casimir forces are discussed. Within the Gaussian approximation we determine the analytic expressions for the associated universal scaling functions and study quantitatively in detail their qualitative features as well as their various limiting behaviors close to the bulk critical point. In addition we consider the effects of time-dependent fields on the fluctuation-induced dynamic Casimir force and determine analytically the corresponding universal scaling functions and their asymptotic behaviors for two specific instances of instantaneous perturbations. The universal aspects of nonlinear relaxation from an initially ordered state are also discussed emphasizing the different crossovers occurring during this evolution. The model considered is relevant to the critical dynamics of actual uniaxial ferromagnetic films with symmetry-preserving conditions at the confining surfaces and for Monte Carlo simulations of spin system with Glauber dynamics and free boundary conditions.     

Higher-order correlation functions of the planar ising model

A method of calculating the asymptotic behaviour of the higher-order correlation functions for large distances is proposed for the planar Ising model in the absence of a magnetic field. The three-point correlation functions composed of a spin operator or of energy-density operators are considered. The asymptotic behaviour of the correlation functions for distances R ? R_c (where R_c is the correlation radius) is determined. It is shown that the asymptotic behaviour of the correlation functions for large distances does not depend on the choice of operators. The asymptotic behaviour of the correlation functions in which two operators are relatively close to one another is considered near the critical point. The results which we obtained are compared with the predictions of the scaling laws and operator algebra.     

Dynamics of Non- interacting System with Long-Range Correlated Quenched Impurities

The theoretic renormalization group approach is applied to the study of the critical behavior of non-interacting system with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ). Totwo-loop order, the asymptotic scaling laws and the critical exponents are studied in the frame of a double (ε, ρ)expansion with ρ of order ε = 4 - d. In d ＜ 4, it is argued that the initial slip exponent θ = 0 together with the dynamicexponent z ＜ 2 is exact in this kind of random system.     

Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics

We investigate the scaling properties of postmortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular, a roughness exponent zeta approximately 0.75 in homogeneous glass and zeta approximately 0.4 in glassy ceramics. The ranges of length scales over which these two scalings are observed are shown to be below and above the size of the process zone, respectively. A model derived from linear elastic fracture mechanics in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect the damage screening occurring for length scales below the size of the process zone.     

Development of particle multiplicity distributions using a general form of the grand canonical partition function

Various phenomenological models of particle multiplicity distributions are discussed using a general form of a unified model which is based on the grand canonical partition function and Feynman's path integral approach to statistical processes. These models can be written as special cases of a more general distribution which has three control parameters which are a,x,z. The relation to these parameters to various physical quantities are discussed. A connection of the parameter a with Fisher's critical exponent τ is developed. Using this grand canonical approach, moments, cumulants and combinants are discussed and a physical interpretation of the combinants are given and their behavior connected to the critical exponent τ. Various physical phenomena such as hierarchical structure, void scaling relations, Koba–Nielson–Olesen or KNO scaling features, clan variables, and branching laws are shown in terms of this general approach. Several of these features which were previously developed in terms of the negative binomial distribution are found to be more general. Both hierarchical structure and void scaling relations depend on the Fisher exponent τ. Applications of our approach to the charged particle multiplicity distribution in jets of L3 and H1 data are given.     

Unstable state dynamics: Treatment of colored noise

The diffusion of a particle set near an unstable point in a bistable potential is considered. The scaling theory of fluctuations proposed originally for onedimensional systems driven by Gaussian white noise is extended to arbitrary dimensions. The merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension. It is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process. The two different methods to include asymptotic fluctuations-which are absent in the usual scaling solution-are shown to be equivalent. An alternate way of including asymptotic fluctuations is attempted by solving the associated Fokker-Planck equation using the Fer formula. The reason for the failure of this method is traced. After this, it is argued that the unified scaling theory should be applicable for treatment of colored noise as well, for the scaling assumption is independent of the statistical property of the driving noise. Explicit Monte Carlo simulation of a model onedimensional system driven by exponentially correlated Gaussian noise is performed and compared with the scaling solution to bolster this point. The agreement is very good.     

A New Treatment for Some Periodic Schr?dinger Operators Ⅱ: The Wave Function

Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schrödinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N=2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended.     

Absence of hyperscaling violations for phase transitions with positive specific heat exponent

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of generalH-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases =3, =5 and =15 where is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.     

Aperiodic extended surface perturbations in the Ising model

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power of the distance l from the free surface with an oscillating amplitude where follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent , the wandering exponent which governs the fluctuation of the sequence and the bulk correlation length exponent . Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the -plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed. Received 1 December 1998     

Quantum critical dynamics simulation of dirty boson systems

Recently, the scaling result z=d for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of z. We study a model of quantum bosons at T=0 with disorder in 2D using highly effective worm Monte?Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are z=1.8±0.05, ν=1.15±0.03, and η=-0.3±0.1, hence suggesting that z=2 is not satisfied.     

A general isomorphism approach to thermodynamic and transport properties of binary fluid mixtures near critical points

A general isomorphism approach to critical phenomena in binary fluid mixtures that may exhibit complex critical-line behavior is developed by relating the two relevant scaling fields to linear combinations of three physical field variables. These physical field variables are related to the temperature and chemical potentials of the two components. The proposed approach includes crossover from vapor-liquid critical behavior to liquid-liquid critical behavior and incorporates also the critical behavior near other special points on critical loci. It is shown that the key factor which determines the apparent behavior of the thermodynamic and transport properties of near-critical mixtures is the shape of the critical locus. The number of system-dependent coefficients that determine the asymptotic critical behavior is elucidated. The choice of zero-points of entropy and energy in binary mixtures is also discussed. The approach provides a powerful tool for predicting thermodynamic and transport properties of fluid mixtures in the critical region.     

Dynamics of critical fluctuations in a binary mixture of limited miscibility under a strong electric field

Experimental investigations of relaxation after switching off the strong electric field in a nitrobenzene-dodecane mixture are presented. Studies were conducted for mixtures of critical and noncritical concentrations using the time-resolved nonlinear dielectric effect. The decays obtained can be portrayed by means of the stretched exponential function with the value of the exponent in agreement with the dynamic droplet model predictions. It has been shown that experimental decays exhibit a universal scaling behavior. The relaxation time (scaling factor) shows a power behavior with the exponent y approximately 1.2 for the critical mixture and y-->1 for the noncritical one. These values are much smaller than theoretically predicted y=1.8-1.9. Based on the assumption that a strong electric field induces in the mixture a quasinematic structure with semiclassical critical properties, a quantitative explanation of this difference is proposed.     

On limit theorems for the bivariate (magnetization,energy) variable at the critical point

The limiting (magnetization, energy) bivariate variable is studied for Ising ferromagnets at the critical point. The factorization property of the limiting bivariate moment generating function is shown to be intimately connected to critical point exponent inequalities and to the behaviour of the scaling limit near and at the critical point. The validity of this can be deduced from the study of the second and the fourth magnetization cumulants at zero external field. The limiting bivariate variable is exactly calculated at the critical point for the Curie-Weiss model (MF) and for the edge of a two-dimensional Ising ferromagnet wrapped on a cylinder. It is shown that the mean field case leads to a non-Gaussian limiting distribution in contradistinction with the particular Ising model we consider for which we obtain a product of two Gaussian probability distributions.     

A JUSTIFICATION FOR THE SCALING OF THE THOM-SYSTEM

In the present paper we discuss the critical behavior of Thornsystem using Catastrophe Theory. The universal critical asymptotic form of the family of free energy functions for Thomsystem with one order parameter and two field parameters is obtained. The expressions of critical exponents, the scaling laws, and the scaling hypotheses are all derived from this universal asymptotic form.