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We have considered a generalized version of a model consisting of self-propelled particles, where the speed is treated as an independent stochastic variable described by a Gamma-distribution, and the motion direction is adjusted in according to two behavioral rules: repulsion and orientation. We used the finite-size scaling theory to estimate the static critical exponents β , γ and ν as a function of the density ρ . We have obtained different values of the critical exponents for different values of the density ρ .

The Kibble–Zurek scaling describes the driven critical dynamics starting with an equilibrium state far away from the critical point. Recently, it has been shown that scaling behaviors also exist when the fluctuation term changes starting near the critical point. In this case, the relevant initial conditions should be included in the scaling theory as additional scaling variables. Here, we study the driven quantum critical dynamics in which a symmetry-breaking field is linearly changed starting from the vicinity of the critical point. We find that, similar to the case of changing the fluctuation term, scaling behaviors in the driven dynamics can be described by the Kibble–Zurek scaling with the initial symmetry-breaking field being included as its additional scaling variable. Both the cases of zero and finite temperatures are considered, and the scaling forms of the order parameter and the entanglement entropy are obtained. We numerically verify the scaling theory by taking the quantum Ising model as an example.

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary., Comment: 21 pages, 4 figures

Critical behavior of double perovskite Y2NiMnO6 near the second-order ferromagnetic transition is studied. Scaling exponents calculated frommodified Arrot plots are confirmed by Kouvel-Fisher method and satisfy the Widom's scaling relation. The exponents do not follow any conventional theoretical models.beta values areconsistent with 3D-Ising model while delta conformsto TCMF and gamma valueclosely relates to the 3D-Heisenberg model. Critical exponents are compared with similar R2NiMnO6 double perovskites which shows that a decrease in size of R ion changes exponents from mean-field to the 3D-Ising model.

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE.

A long-standing issue in a three-dimensional Ising spin glass (3DISG) is to settle the critical exponents. Numerical simulations and a conventional finite-size-scaling analysis have given controversial estimates, in particular, for the value of thermal exponent ν. Here a recently proposed extended scaling scheme is reviewed and illustrated for the canonical two-dimensional Ising ferromagnet. By applying the extended scaling to our extensive Monte Carlo data for 3DISG, consistent estimates of critical parameters are obtained in contrast to previous analyses.

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal stochastic Loewner evolution (SLE6), the full scaling limit of critical cluster boundaries, and near-critical scaling limits.

In this work we study the critical behavior of a model that simulates the propagation of an epidemic process over a population. We simulate the model on two-dimensional finite lattices to determine the critical density of the diffusive population. A finite size scaling analysis is employed to determine the order parameter and correlation length critical exponents.

Purpose – The paper presents an application of the scaling theory in a description of energy losses in soft magnetic materials in order to improve an agreement between measurements and theoretical models.Design/methodology/approach – The scaling theory allows the description of energy losses by a generalized homogenous function, which depends on scaling exponents α, β and amplitudes Γ(n). The values of the scaling exponents and amplitudes were estimated on the basis of measurement data of total energy losses.Findings – The main findings of the paper are: the linear relationships between the scaling exponents α and β, the data collapse of energy losses and the scaling laws for asymptotic exponents of energy losses derivatives.Research limitations/implications – The origin of the data collapse and the relationship between the scaling exponents will be the subject of further research with the aid of renormalization group methodPractical implications – The paper could be useful both for device designers and r...

We have extended dynamic scaling hypothesis to describe the wide range of experiments on growing rough interfaces. An alternative approach of original Family–Vicsek scaling hypothesis was not successfully applied to the growing interfaces of paper wetting in that we obtained unphysical values of roughness exponents. Since very nice data collapse suggested the existence of some kind of dynamic scaling, we have tried to extend the dynamic scaling hypothesis. It was found that the extended dynamic scaling hypothesis can be nicely applied to the growing rough interfaces in paper wetting, yielding values of roughness and growth exponents α≃0.73, β≃0.60, respectively.

Using a finite-size phenomenological theory we investigate the behavior of the Baxter–Wu model for both first- and second-order transitions. In order to distinguish between the two kinds of transition we study the finite-size scaling behavior of the order parameter and the susceptibility of the model. At the critical temperature the known critical exponents of the system govern the scaling behavior, while below the critical temperature the scaling exponents are the dimensionality of the system. The presented scaling theory for the Baxter–Wu model is an appropriate generalization of the corresponding theory of Binder and Landau for the Ising model [Binder and Landau, Phys. Rev. B 30 (1984) 1477]. We perform numerical simulations using standard Monte Carlo techniques and we find that our results are in good agreement with theoretical predictions.

A systematic way for deriving fundamental earthquake scaling relationships in their size-frequency distribution and high-frequency spectrum falloff based on the renormalization group method is presented for the model earthquakes. It is known that faults populations are broadly scale-invariant over several orders of magnitude. Applying the renormalization group method to the generalized Burrige–Knopoff earthquake model, we obtain the self-similar solution from which the scaling exponents of the size-frequency distribution and spectrum falloff are obtained. The obtained exponents are in agreement with the known values and the relation between these scaling exponents helps to bridge the gap between statistic and dynamical scalings. The present paper permits us to study the self-similar behavior and helps in understanding the basic ingredients underlying earthquake scaling.

A novel method to estimate the critical point and critical exponents of physical models from numerical studies is presented. The method utilizes linear approximation to compute the values of the characteristic variables in the near vicinity of the critical point from numerical results obtained for only one point in that vicinity. The method is applied to two models: Two-dimensional directed percolation, and one-dimensional reaction-diffusion model. In both cases the critical point and critical exponents are determined with higher accuracy than achieved in former studies. In the reaction-diffusion model, the results strongly suggest simple rational values of 1/3, 1/12, etc., for the characteristic exponents.

We propose to combine the nonlinear scaling fields associated with the high-temperature (HT) fixed point, with those associated with the unstable fixed point, in order to calculate the susceptibility and other thermodynamic quantities. The general strategy relies on simple linear relations between the HT scaling fields and the thermodynamic quantities, and the estimation of RG invariants formed out of the two sets of scaling fields. This estimation requires convergent expansions in overlapping domains. If, in addition, the initial values of the scaling fields associated with the unstable fixed point can be calculated from the temperature and the parameters appearing in the microscopic Hamiltonian, one can estimate the critical amplitudes. This strategy has been developed using Dyson's hierarchical model where all the steps can be approximately implemented with good accuracy. We show numerically that for this model (and a simplified version of it), the required overlap apparently occurs, allowing an accurate determination of the critical amplitudes.

Nature Communications 7: Article number: 13611 (2016); Published 5 December 2016; Updated 17 January 2017 The original version of this Article contained a typographical error in the spelling of the author S.A. Cannas, which was incorrectly given as S. Cannas. This has now been corrected in both the PDF and HTML versions of the Article.

Is the brain really operating at a critical point? We study the non-equilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasi-critical dynamics on the Widom line where the correlation length is maximal. We simulate the network and introduce an analytical mean-field approximation, characterize the non-equilibrium phase transition, and present a non-equilibrium phase diagram, which shows that in addition to an ordered and disordered phase, the system exhibits a quasiperiodic phase corresponding to synchronous activity in simulations which may be related to the pathological synchronization associated with epilepsy., Submitted to Physical Review Letters on March 29, 2014. Transferred to Physical Review E on August 4, 2014

We investigate a method to obtain critical exponents using standard numerical simulation techniques for spin systems. We show that by analyzing the inverse of the logarithmic derivative of the susceptibility χ with respect to temperature the accuracy in determining γ can be increased significantly. The method is tested on the two- and three-dimensional Ising models but this alternative approach is thought to be most useful for spin systems on non-periodic structures like quasi-crystals and fractal lattices where the finite size scaling method is difficult to apply.

The six-dimensional Ising model is simulated on the Creutz cellular automaton using the finite-size lattices with the linear dimension 4 less than or equal to L less than or equal to 10. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 3.00(12) and 1.47(10) using 6 less than or equal to L less than or equal to 10, respectively, which are in very good agreement with the theoretical predictions of 6/2 and a. The critical temperature for the infinite lattice is found to be 10.835(5) using 4 less than or equal to L less than or equal to 10 which is also in very good agreement with the precise results. The finite-size scaling relation fur the magnetic susceptibility at the infinite-lattice critical temperature is also valid for the maxima of the magnetic susceptibilities of the finite-size lattices. The finite-size scaling plots of the magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice critical temperature. (C) 2000 Elsevier Science B.V. All rights reserved.

The bond-percolation process is studied on periodic planar random lattices and their duals. The thresholds and critical exponents of the percolation transition are determined. The scaling functions of the percolating probability, the existence probability of the appearance of percolating clusters, and the mean cluster size are also calculated. The simulation result of the percolation threshold is pc50.333360.0001 for planar random lattices, and 0.667060.0001 for the duals of planar random lattices. We conjecture that the exact value of pc is 1/3 for a planar random lattice and 2/3 for the dual of a planar random lattice. By taking possible errors into account, the results of our critical exponents agree with the values given by the universality hypothesis. By properly adjusting the metric factors on random lattices and their duals, we demonstrate explicitly that the idea of a universal scaling function with nonuniversal metric factors in the finite-size scaling theory can be extended to random lattices and their duals for the existence probability, the percolating probability, and the mean cluster size. @S1063-651X~99!00112-9#

The four-dimensional Ising model is simulated on the Creutz cellular automaton with increased precision. The data are analyzed according to the finite-size scaling relations available. The precision of the critical values related to magnetic susceptibility is improved by one digit, but in order to reach to the same precision for those related to the specific heat more simulation runs at the critical temperatures of the finite-size lattices are required.

We consider a mean-field systematic approach to study the critical behavior of cooperative models. The sequence of approximate mean-field results are analyzed by the introduction of a scaling hypothesis from which we can obtain the critical parameter as well as the nonclassical critical exponents. The method is applied to the Domany–Kinzel probabilistic cellular automaton and to the two-dimensional Glauber–Ising model.

The persistence of scaling laws in the evolution of the automaton Game of Life from a correlated configuration is investigated. It is found that even for a highly correlated initial state, the temporal evolution is able to preserve scaling relations as observed for uncorrelated states, although with possibly discordant critical exponents. The difference can be accounted for by introducing a stochastic component in the transition rules, which at the same time establishes a connection with the possible convergence of Life to a self-organized critical state.

A method is proposed to determine the dynamical critical exponent. The method is based on the scaling for dynamical hysteresis resulted from a linearly swept field. We prove that in model $A$ dynamical hysteresis scaling at critical temperature is universal. The nearest-neighbor Ising models are used to demonstrate such concepts and the dynamical critical exponents can be determined accurately. We also propose a universal relation between static and dynamical critical exponents in Ising class in single-spin-flip dynamics.

In this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents ζn of the nth order correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζn. This hierarchy was discussed in detail in a recent publication [V.S. L’vov and I. Procaccia, Physica A (1998), in press, chao-dyn/9707015]. The scaling exponents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order non-trivial closures of this infinite set of equations, and prove that these closures lead to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier–Stokes equations. Nevertheless, they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Holder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L.

The Kolmogorov equation, which is an exact relationship between the second-order structure function and the third-order structure function , is applied to study the relative scaling of against according to the extended self-similarity (ESS) method. It is found that the relative ESS scaling exponent is greater than the real (or theoretical) inertial range scaling exponent of in the case of normal scaling as well as anomalous scaling : when and when . Therefore, the experimental and numerical results favours the Kolmogorov 2/3 law rather than anomalous scaling . Previously the results were interpreted as clear evidence of anomalous scaling based upon the assumption that .

We explore the intersection properties of stars of random and self-avoiding walks. We show how the corresponding scaling exponents govern the scaling behavior of copolymer networks in solution. We derive and calculate these exponents from a renormalization group analysis of a corresponding Edwards Hamiltonian. Our third-order spectrum of exponents calculated by field-theoretic renormalization shows remarkable features: All exponents are scaling dimensions of composite power of field operators, convexity of the spectrum allows for a multifractal interpretation, and the 2D limit has no simple Kac formula like structure.

We use heat bath dynamics to evaluate the dynamic critical exponent z and the dynamic finite-size scaling function of an Ising model on square, planar triangular, and honeycomb lattices. We find convincing evidence that z is universal and, by choosing an aspect ratio and a nonuniversal metric factor for the scaled time of each lattice, we can obtain a universal dynamic finite-size scaling function for the Ising model on the planar lattices. Our results suggest many interesting problems for further research. @S1063-651X~97!15408-3#

We present a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with non-identical natural frequencies in terms of the Kuramoto model. This provides a comprehensive view of phase synchronization. In particular, we extend finite-size scaling (FSS) in the steady state to dynamics, determine critical exponents, and find the critical coupling strength. The dynamic scaling approach enables us to measure not only the FSS exponent associated with the correlation volume in finite systems but also thermodynamic critical exponents. Based on the extended FSS theory, we also discuss how the sampling of natural frequencies and thermal noise affect dynamic scaling, which is numerically confirmed., 8 pages, 8 figures (18 eps files), 1 table; publised version

It is shown that the recently proposed finite size scaling renormalization group, when using systems infinite in one dimension and finite in the others, is equivalent to the Nightingale (correlation length) phenomenological renormalization. The equivalence, however, is concerned only with the critical coupling and thermal critical exponent; the finite size scaling renormalization group approach provides other exponents in a more precise and elegant fashion through the flux diagram of the recursion relations in the space spanned by the parameters of the Hamiltonian. In addition, a new set of magnetic exponents, which are so accurate as the thermal ones, can now be obtained in an easier way.

Physicists have often observed a scaling behaviour of the main physical quantities during experiments on systems exhibiting a phase transition. The main assumption of a scaling theory is that these characteristic quantities are self-similar functions of the independent variables of the phenomenon and, therefore, such a scaling can be interpreted be means of power-laws. Since a characteristic feature of phase transitions is a catastrophic change of the macroscopic parameters of the system undergoing a continuous variation in the system state variables, the phenomenon of fracture of disordered materials can be set into the wide framework of critical phenomena. In this paper new mechanical properties are defined, with non integer physical dimensions depending on the scaling exponents of the phenomenon (i.e. the fractal dimension of the damaged microstructure, or the exponent of a power-constitutive relation), which turn out to be scale-invariant material constants. This represents the so-called renormalization procedure, already proposed in the statistical physics of random process.

We present a new technique to describe the abnormal behavior of certain fluctuation observables in the critical regime of quantum statistical systems which undergo a phase transition. The idea is to rescale the local fluctuation operators by a relevant external parameter of the system, in addition to the usual scaling with the inverse square root of the volume. The scaling indices used in this scaling procedure are directly related to the critical exponents. Furthermore, it is explained that this new method of scaling preserves the CCR structure of the algebra of macroscopic fluctuations. Finally, scaling indices are computed for the relevant microscopic observables at all temperatures in a mean field approximation for a quantum anharmonic crystal. These indices yield the same critical exponents as predicted by mean field theory.

The Privman-Fisher universal amplitude, characterizing the free energy of a finite-sized system at its critical point, is identified as the critical counterpart of the 'number of stable phases' on the coexistence curve. Its value is shown to be determined by the critical-point form of the order parameter distribution, directly accessible to Monte Carlo studies.

We investigate families of generalized mean--field theories that can be formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians describing mutually non--interacting subsystems of increasing size, the thermodynamics of these mean--field type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean--field critical points. Finite--size scaling analysis of this mean--field critical behaviour allows to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki, which is thus given a transparent interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls--Bogoliubov inequality. In the case of the 2--$d$ Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity of thermodynamic functions., Comment: 17 pages, latex, 3 figures appended as uuencoded compressed tar file

To determine the universality class of critical phenomena, we propose a method of statistical inference in the scaling analysis of critical phenomena. The method is based on Bayesian statistics, most specifically, the Gaussian process regression. It assumes only the smoothness of a scaling function, and it does not need a form. We demonstrate this method for the finite-size scaling analysis of the Ising models on square and triangular lattices. Near the critical point, the method is comparable in accuracy to the least-square method. In addition, it works well for data to which we cannot apply the least-square method with a polynomial of low degree. By comparing the data on triangular lattices with the scaling function inferred from the data on square lattices, we confirm the universality of the finite-size scaling function of the two-dimensional Ising model., Comment: 7 pages, 8 figures, and 1 table

We analyze the entropic equation of state for a many-particle interacting system in a scale-free network. The analysis is performed in terms of scaling functions, which are of fundamental interest in the theory of critical phenomena and have previously been theoretically and experimentally explored in the context of various magnetic, fluid, and superconducting systems in two and three dimensions. Here, we obtain general scaling functions for the entropy, the constant-field heat capacity, and the isothermal magnetocaloric coefficient near the critical point in uncorrelated scale-free networks, where the node-degree distribution exponent $\ensuremath{\lambda}$ appears to be a global variable and plays a crucial role, similar to the dimensionality $d$ for systems on lattices. This extends the principle of universality to systems on scale-free networks and allows quantification of the impact of fluctuations in the network structure on critical behavior.

The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10-6. Furthermore we estimate the critical density of the hard-square model as rho c=0.367 743 000 (5).

The scaling of the point-point correlation function of DLA has shown itself to be more complex and challenging than had at first been imagined. This paper determines from the results of previous studies that a more general scaling rule must be applied, at least robust enough to allow different moments of the point-point correlation function to scale non-trivially, and for the scale length to scale non-trivially. In this sense, it is seen that the scaling problem is analogous to that of multifractal scaling. The authors present a group-theoretic derivation of the most general scaling transformations possible, and show that it is adequately robust to explain the non-trivial scaling observed in the moments of the point-point correlation function.

We apply finite-time scaling to the $q$-state Potts model with $q=3$ and 4 on two-dimensional lattices to determine its critical properties. This consists in applying to the model a linearly varying external field that couples to one of its $q$ states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields. From the finite-time scaling of the order parameter, the coercivity, and the hysteresis area and its derivative, we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature. The static critical exponents obtained in general and the magnetic exponent $\ensuremath{\delta}$ in particular agree reasonably with the conjectured ones. The dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short-time dynamic results for $q=4$. Our results also suggest an alternative way to characterize the weak universality.

We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.

The finite-size scaling (FSS) theory for continuous phase transitions has been useful in determining the critical behavior from the size dependent behaviors of thermodynamic quantities. When the phase transition is discontinuous, however, FSS approach has not been well established yet. Here, we develop a FSS theory for the explosive percolation transition arising in the Erd\H{o}s and R\'enyi model under the Achlioptas process. A scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point $t_c$ diverges with system size in a power-law manner, which is different from the conventional one based on the divergence of the correlation length at $t_c$. We show that the susceptibility is also described in the same scaling form. Numerical simulation data for different system sizes are well collapsed on the respective scaling functions., Comment: 5 pages, 5 figures

A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f→0 provided that the time scales of free growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, there is evidence that the forest density in the critical state assumes its minimum possible value, i.e. that energy dissipation is maximum.

The present review is devoted to the fundamental problems of finite-size scaling due to the presence of long-range interactions. The attention is focused on the precise formulation of critical finite-size scaling in the case that the bulk correlation length is identically infinite. The hypotheses, based on the notion of a finite-size scaling length, are formulated in a way which treats the cases of short- and long-range interactions on equal grounds, as well as the standard and modified (above the upper critical dimensionality) finite-size scaling. The general conjectures are tested on the exactly solvable mean spherical model with power-law interaction. Special attention is paid to the adequate mathematical techniques. The formulation of finite-size scaling at first-order phase transitions is also discussed, since the finite-size scaling length enters into the definition of the relevant scaled external field variable provided the system has d ′>0 infinite dimensions. The review closes with a summary and short discussion of some open problems.

A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f\ensuremath{\rightarrow}0 provided that the time scales of tree growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, we show that the forest density in the critical state assumes its minimum possible value, i.e., that energy dissipation is maximum.

We present a Monte Carlo study of the one and the two component / gf 4 model on the cubic lattice in three dimensions. Using a finite size scaling method we determine the value of the coupling λ at which leading corrections to scaling vanish. Using this result we obtain precise estimates for critical exponents.

In this work, we study the critical behavior of a one-dimensional model that mimics the propagation of an epidemic process mediated by a density of diffusive individuals which can infect a static population upon contact. We simulate the above model on linear chains to determine the critical density of the diffusive population, above which the system achieves a statistically stationary active state, as a function of two relevant parameters related to the average lifetimes of the diffusive and nondiffusive populations. A finite-size scaling analysis is employed to determine the order parameter and correlation length critical exponents. For high-recovery rates, the critical exponents are compatible with the usual directed percolation universality class. However, in the opposite regime of low-recovery rates, the diffusion is a relevant mechanism responsible for the propagation of the disease and the absorbing state phase transition is governed by a distinct set of critical exponents.

We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte-Carlo simulations for times up to $10^{10}$ and system sizes up to $8000 \times 8000$ sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model., 13 pages, 12 eps figures, final version as published

We study three sand pile automaton models namely, the critical height, critical slope and the critical Laplacian models in two dimensions, in which the stability criterion of the sand columns depend on the zeroth, first and the second derivatives of the sand height function. We carried out simulations on system sizes up to 2048 × 2048 and up to 108 avalanches were generated. The exponents of the critical height model were calculated by taking into account the strong corrections to scaling. In order to determine the exponents of the critical Laplacian model accurately we introduced a height restriction in the toppling criterion that maintains universality but accelerates convergence to the steady state by orders of magnitude. We see clear scaling in the critical height and the critical Laplacian models and find that they belong to different universality classes. However, we do not find any scaling in the critical slope model.

The properties of an infinite system at a continuous phase transition are charac-terised by non-trivial critical exponents. These non-trivial exponents are related to scalingrelations of the thermodynamic potential. The scaling properties of the singular part of thespecific entropy of infinite systems are deduced starting from the well-established scaling re-lations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling aresuppressed in the microcanonical ensemble compared to the corresponding corrections in thecanonical ensemble.Keywords: Critical phenomena, microcanonical entropy, scaling relations, Gaussian andspherical model1. IntroductionIn a mathematically idealised way continuous phase transitions are described in infinite systems al-though physical systems in nature or in a computer experiment are finite. The singular behaviour ofphysical quantities such as the response functions is related to scaling properties of the thermodynamicpotential which is used for the formulation [1, 2]. Normally the Gibbs free energy is chosen for thedescription although other potentials also contain the full information about an equilibrium phase transi-tion. The Hankey-Stanley theorem relates the scaling properties of the Gibbs free energy to the scalingproperties of any other energy-like potential that is obtained from the Gibbs free energy by a Legendretransformation [3]. However, the thermodynamic properties can also be deduced from entropy-like po-tentials, the Massieu-Planck functions, that are related to energy-like potentials by partial inversions [4].In this paper the scaling relation of the Gibbs free energy is translated to the entropy which is related tothe degeneracy of configurations with respect to macroscopic quantities. The considerations are concen