Coherent anomalies and an asymptotic method in cooperative phenomena

A new general method is proposed to study the asymptotic behavior of cooperative systems. It is based on the appearance of a “coherent anomaly” in mean-field-type approximations for cooperative systems and is inspired by Fisher's finite-size scaling theory. It is promising for calculating analytically non-classical scaling exponents in cooperative phenomena. Even the combination of the Weiss and Bethe approximations gives rather good values of static and dynamic critical exponents.     

Critical behavior in random-spin systems with long-range interactions

Effects of the potential range of the interaction to critical behaviors of quenched random-spin systems are investigated in the limit M → 0 of the MN-component models by means of the renormalization-group theories. As static critical phenomena the stability of the fixed points is investigated and the critical exponents (η, γ, , crossover index) and the equation of state are derived. These phenomena are different from those in pure systems, for the positive specific heat exponent of the pure Heisenberg system.     

Computer simulation studies of magnetic phase transitions

The advancements which have been made in the use of computer simulations to study magnetic-phase transitions and critical phenomena are reviewed. We describe how the use of a combination of sophisticated Monte Carlo simulation algorithms and reweighting (histogram) techniques have allowed the determination of the static critical behavior with unprecedented precision. The study of “dynamic” critical behavior in simple spin models by both Monte Carlo and spin dynamics methods is also reviewed. Recent estimates for dynamic critical exponents are given including those for true dynamics.     

TWO SCALING RELATIONS BETWEEN STATIC AND DYNAMICAL THRESHOLD EXPONENTS AND AMPLITUDES

Dynamical threshold phenomena in non-equili-brium systems satisfying the potential condition are discussed. Two scaling relations between the static and dynamical threshold exponents and amplitudes are obtained.     

Random walk on topologically biased anisotropic percolation clusters and transport properties

Unbiased random walks are performed on topologically biased anisotropic percolation clusters (APC). Topologically biased APCs are generated using suitable anisotropic percolation models. New walk dimensions are found to characterize the anisotropic behaviour of the unbiased random walk on the biased topology. Critical properties of electro and magneto conductivities are characterized estimating respective dynamical critical exponents. A dynamical scaling theory relating dynamical and static critical exponents has been developed. The dynamical critical exponents satisfy the scaling relations within error bar.     

Self-Similarity Breaking: Anomalous Nonequilibrium Finite-Size Scaling and Finite-Time Scaling

Symmetry breaking plays a pivotal role in modern physics.Although self-similarity is also a symmetry,and appears ubiquitously in nature,a fundamental question arises as to whether self-similarity breaking makes sense or not.Here,by identifying an important type of critical fluctuation,dubbed‘phases fluctuations’,and comparing the numerical results for those with self-similarity and those lacking self-similarity with respect to phases fluctuations,we show that self-similarity can indeed be broken,with significant consequences,at least in nonequilibrium situations.We find that the breaking of self-similarity results in new critical exponents,giving rise to a violation of the well-known finite-size scaling,or the less well-known finite-time scaling,and different leading exponents in either the ordered or the disordered phases of the paradigmatic Ising model on two-or three-dimensional finite lattices,when subject to the simplest nonequilibrium driving of linear heating or cooling through its critical point.This is in stark contrast to identical exponents and different amplitudes in usual critical phenomena.Our results demonstrate how surprising driven nonequilibrium critical phenomena can be.The application of this theory to other classical and quantum phase transitions is also anticipated.     

The critical exponent of nuclear fragmentation

Nuclei colliding at energies in the MeV’s break into fragments in a process that resembles a liquid-to-gas phase transition of the excited nuclear matter. If this is the case, phase changes occurring near the critical point should yield a “droplet” mass distribution of the form ≈A ^?T, with T (a critical exponent universal to many processes) within 2≤T≤3. This critical phenomenon, however, can be obscured by the finiteness in space of the nuclei and in time of the reaction. With this in mind, this work studies the possibility of having critical phenomena in small “static” systems (using percolation of cubic and spherical grids), and on small “dynamic” systems (using molecular dynamics simulations of nuclear collisions in two and three dimensions). This is done investigating the mass distributions produced by these models and extracting values of critical exponents. The specific conclusion is that the obtained values of T are within the range expected for critical phenomena, i.e. around 2.3, and the grand conclusion is that phase changes and critical phenomena appear to be possible in small and fast breaking systems, such as in collisions between heavy ions.     

Dynamic critical behavior of semi-infinite systems: the study of the model C

The dynamic critical behavior of semi-infinite model C near the special and ordinary transitions is investigated using field theoretic renormalization-group approach. It is shown that the dynamic surface quantities have different critical behavior aginst their bulk analogues and their scaling laws can be expressed entirely in terms of static (bulk and surface) exponents and dynamic exponents. It is found that at the critical point the surface transport coefficient reaches a finite value via a cusplike singularity and the surface-bulk transport coefficient diverges, but the bulk transport coefficient remains finite as that of the infinite model C.     

Three fundamental problems in ferroelectricity

We review briefly the phenomena of optical bistability in ferroelectrics, the reversible oxidation-reduction processes at electrode interfaces in ferroelectric thin films, and the question of critical exponents in ferroelectrics near tricritical points.     

PHASE TRANSITION ON HIERARCHICAL LATTICES WITH ADDITIONAL DILUTED BONDS

The critical phenomena on hierarchical lattice with additional diluted bonds are investigated for both thermal and geometrical Phase transitions. It is shown that the diluted bonds reinforce the phase transitions but do not change the critical exponents except that they are fully occupied.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/r^d+δ (δ>0) on d-dimensional hypercubic lattices (d=1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z=2 if δ>2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0<δ<2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

Phase transitions in ferromagnets and singularities

This paper is devoted to the investigation of critical phenomena from the mathematical point of view. An interesting question in this field is universality (i.e. independence of the type of material and, to some extent, of the type of process) of the so-called critical exponents. We try to show that one can compute the critical exponents using structural stability arguments. The universality is then implied automatically as a consequence of similarity of models and of the structure of stable singularities. The considerations lead to some new results: information about the generic shape of hysteresis loops and universal form of the equation of state in the critical region.     

Critical phenomena for a triangular spin lattice I: Triplet interactions

Critical phenomena in an Ising-like spin system on a triangular lattice are studied by means of the renormalization theory. The critical indices of the critical point due to triplet interactions are determined approximately, in fair agreement with the exponents, known from the exact solution due to Baxter and Wu. The cross-over from triplet critical behavior to ferromagnetic (nearest neighbor) critical behavior is calculated.     

Critical Relaxation of a Three-Dimensional Fully Frustrated Ising Model

Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.     

Bond dilution in the 3D Ising model: a Monte Carlo study

We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.Received: 24 February 2004, Published online: 28 May 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 64.60.Fr Equilibrium properties near critical points, critical exponents - 75.10.Hk Classical spin models     

100Rh pac study of the critical region of Co: The lessons learned

Observations of hyperfine fields over 295<T<1330 K and relaxation spectra above T_c demonstrate the feasibility of measuring static and dynamic critical exponents with sources of¹⁰⁰PdCo. These PAC experiments promise to clarify current discrepancies between measured exponents and theory in Co once source contamination problems are resolved.     

Computer simulation of the critical behavior of 3D disordered ising model

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.     

Master equation approach to the dynamical critical behaviour of the Schlögl model

Starting from the multivariate master equation we construct a reduced model that gives the critical theory. The relation to the static problem is studied using the minimal subtraction technique and the critical exponents are calculated up to O(?²).     

On self-organised criticality in one dimension

In critical phenomena, many of the characteristic features encountered in higher dimensions such as scaling, data collapse and associated critical exponents are also present in one dimension. Likewise for systems displaying self-organised criticality. We show that the one-dimensional Bak–Tang–Wiesenfeld sandpile model, although trivial, does indeed fall into the general framework of self-organised criticality. We also investigate the Oslo ricepile model, driven by adding slope units at the boundary or in the bulk. We determine the critical exponents by measuring the scaling of the kth moment of the avalanche size probability with system size. The avalanche size exponent depends on the type of drive but the avalanche dimension remains constant.     

Critical behavior of anisotropic cubic systems with long- and short-range interactions

Critical phenomena in anisotropic cubic N-component spin systems with long- and short-range interactions are investigated and discussed in the regions of weakly long-range, intermediate-range, range, and the long-range potentials. The expressions for the eigenvalues and the critical exponents (n,γ and crossover exponents) in these three regions are derived and their stability is discussed. These results of the systems are compared with those of the same isotropic systems.