Universal Finite-size Scaling Functions Research Articles

Universal finite-size scaling function for coarsening in the Potts model with conserved dynamics

We study kinetics of phase segregation in multicomponent mixtures via Monte Carlo simulations of the q-state Potts model, in two spatial dimensions, for 2 ≤ q ≤ 20. The associated growth of domains in finite boxes, irrespective of q and temperature, can be described by a single universal finite-size scaling function, with only the introduction of a nonuniversal metric factor in the scaling variable. Our results show that although the scaling function is independent of the type of transition, the q-dependence of the metric factor hints to a crossover at q = 5 where the type of transition in the model changes from second to first order.

Universal finite-size scaling function for kinetics of phase separation in mixtures with varying number of components

From Kawasaki-exchange Monte Carlo simulations of the $q$-state Potts model, we present results for the kinetics of phase separation in multicomponent mixtures, for $q\ensuremath{\le}10$, in space dimension $d=2$. A particular focus has been on the quantification of finite-size scaling functions for various values of $q$ and quench depths. For a range of final quench temperatures, our analyses, via finite-size scaling and other state-of-the-art methods, show that the growth follows the Lifshitz-Slyozov behavior, expected for a diffusive mechanism, irrespective of the number of components. We show that the growth for different $q$ values and quench temperatures, in finite systems, can be described by a universal scaling function with a nonuniversal metric factor, originating from the differences in the amplitudes. We also demonstrate the morphological and kinetic equivalence between a $q$-component equal proportion mixture and an off-critical binary mixture, in the framework of the Ising model, with relative concentration of the minority component in the latter being ${x}_{c}=1/q$.

Coarsening and aging of lattice polymers: Influence of bond fluctuations.

We present results for the nonequilibrium dynamics of collapse for a model flexible homopolymer on simple cubic lattices with fixed and fluctuating bonds between the monomers. Results from our Monte Carlo simulations show that, phenomenologically, the sequence of events observed during the collapse are independent of the bond criterion. While the growth of the clusters (of monomers) at different temperatures exhibits a non-universal power-law behavior when the bonds are fixed, the introduction of fluctuations in the bonds by considering the existence of diagonal bonds produces a temperature independent growth, which can be described by a universal nonequilibrium finite-size scaling function with a non-universal metric factor. We also examine the related aging phenomenon, probed by a suitable two-time density-density autocorrelation function showing a simple power-law scaling with respect to the growing cluster size. Unlike the cluster-growth exponent αc, the nonequilibrium autocorrelation exponent λC governing the aging during the collapse, however, is independent of the bond type and strictly follows the bounds proposed by Majumder and Janke [Phys. Rev. E 93, 032506 (2016)] at all temperatures.

Kinetics of polymer collapse: effect of temperature on cluster growth and aging.

Using state of the art Monte Carlo simulations of a bead-spring model we investigate both the equilibrium and the nonequilibrium behavior of the homopolymer collapse. The equilibrium properties obtained via multicanonical sampling recover the well-known finite-size scaling behavior of collapse for our model polymer. For the nonequilibrium dynamics we study the collapse by quenching the homopolymer from an expanded coiled state into the globular phase. The sequence of events observed during the collapse is independent of the quench depth. In particular, we focus on finding out universal scaling behaviors related to the growth or coarsening of clusters of monomers, by drawing phenomenological analogies with ordering kinetics. We distinguish the cluster coarsening stage from the initial stage of primary cluster formation. By successful application of a nonequilibrium finite-size scaling analysis we show that at all quench temperatures, during the coarsening stage, the cluster growth is roughly linear and can be characterised by a universal finite-size scaling function. In addition, we provide evidence of aging by constructing a suitable autocorrelation function and its corresponding dynamical power-law scaling with respect to the growing cluster sizes. The predicted theoretical bound for the exponent governing such scaling is strictly obeyed by the numerical data irrespective of the quench temperature. The results and methods presented here in general should find application in similar phenomena such as the collapse of a protein molecule preceding its folding.

Influence of the aspect ratio and boundary conditions on universal finite-size scaling functions in the athermal metastable two-dimensional random field Ising model.

This work studies universal finite size scaling functions for the number of one-dimensional spanning avalanches in a two-dimensional (2D) disordered system with boundary conditions of different nature and different aspect ratios. To this end, we will consider the 2D random field Ising model at T=0 driven by the external field H with athermal dynamics implemented with periodic and forced boundary conditions. We have chosen a convenient scaling variable z that accounts for the deformation of the distance to the critical point caused by the aspect ratio. In addition, assuming that the dependence of the finite size scaling functions on the aspect ratio can be accounted for by an additional multiplicative factor, we have been able to collapse data for different system sizes, different aspect ratios, and different types of the boundary conditions into a single scaling function Q̂.

Thermodynamic Casimir effect in films: the exchange cluster algorithm.

We study the thermodynamic Casimir force for films with various types of boundary conditions and the bulk universality class of the three-dimensional Ising model. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. In particular, we employ the exchange or geometric cluster cluster algorithm [Heringa and Blöte, Phys. Rev. E 57, 4976 (1998)]. In a previous work, we demonstrated that this algorithm allows us to compute the thermodynamic Casimir force for the plate-sphere geometry efficiently. It turns out that also for the film geometry a substantial reduction of the statistical error can achieved. Concerning physics, we focus on (O,O) boundary conditions, where O denotes the ordinary surface transition. These are implemented by free boundary conditions on both sides of the film. Films with such boundary conditions undergo a phase transition in the universality class of the two-dimensional Ising model. We determine the inverse transition temperature for a large range of thicknesses L(0) of the film and study the scaling of this temperature with L(0). In the neighborhood of the transition, the thermodynamic Casimir force is affected by finite size effects, where finite size refers to a finite transversal extension L of the film. We demonstrate that these finite size effects can be computed by using the universal finite size scaling function of the free energy of the two-dimensional Ising model.

Finite size dependence of scaling functions of the three-dimensional O(4) model in an external field

We calculate universal finite size scaling functions for the order parameter and the longitudinal susceptibility of the three-dimensional O(4) model. The phase transition of this model is supposed to be in the same universality class as the chiral transition of two-flavor QCD. The scaling functions serve as a testing device for QCD simulations on small lattices, where, for example, pseudocritical temperatures are difficult to determine. In addition, we have improved the infinite volume limit parametrization of the scaling functions by using newly generated high statistics data for the 3d O(4) model in the high temperature region on an L=120 lattice.

Historical Review on Analytic, Monte Carlo, and Renormalization Group Approaches to Critical Phenomena of Some Lattice Models

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

Finite size scaling in the local abundances of geographic populations

We analyzed the statistical distribution of intra-specific local abundances for a set North American breeding bird species. We constructed frequency plots for every species and found that they showed long-tail power-law behavior, truncated at an upper abundance cut-off value. Based on finite size scaling arguments, we investigated whether frequency curves may be considered scaled copies of each other. Data collapse was possible after taking powers of the total abundance of each species, in order to correct deviations from the underlying universal finite size scaling function (UFSS). The UFSS power law exponent oscillated in time within the regime of unbounded variance, which is consistent with the wild fluctuations that characterize ecological phenomena. We speculate that our results may eventually be linked to other law-like macroecological phenomena, such as energetic constraints reported in allometric scaling.

Universality and universal finite-size scaling functions in four-dimensional Ising spin glasses

We study the four-dimensional Ising spin glass with Gaussian and bond-diluted bimodal distributed interactions via large-scale Monte Carlo simulations and show via an extensive finite-size scaling analysis that four-dimensional Ising spin glasses obey universality.

Thermodynamic Casimir Effect inHe4Films nearTλ: Monte Carlo Results

The universal finite-size scaling function of the critical Casimir force for the three dimensional $XY$ universality class with Dirichlet boundary conditions is determined using Monte Carlo simulations. The results are in excellent agreement with recent experiments on $^{4}\mathrm{He}$ Films at the superfluid transition and with available theoretical predictions.

Critical behavior of the three-dimensional bond-diluted Ising spin glass: Finite-size scaling functions and universality

We study the three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution ${p}^{*}$, at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.

Anisotropy and restricted universality of critical phenomena

We study the 4 lattice model with anisotropic short-range interactions. We show that the nonuniversal finite-size scaling functions of the anisotropic 4 model in a d-dimensional rectangular geometry are determined by the universal finite-size scaling functions of the isotropic 4 theory in a d-dimensional parallelepiped. Predictions are made for the Binder cumulant. In the bulk limit two-scale factor universality is absent in those universal critical-point amplitude relations that involve the correlation length.

Critical Casimir forces forO(n)systems with long-range interaction in the spherical limit

We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the d -dimensional spherical model with a power law long-ranged interaction decaying at large distances r as r(-d-sigma) , where sigma<d<2sigma and 0<sigma< or =2 . For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature T(c) , as well as for T> T(c) and T< T(c) . The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and subcritical region the force is of the order of L(-d) , for T> T(c) it decays as L(-d-sigma) , where L is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when d-1<sigma , as well as the case with d-1>sigma , when one observes a dimensional crossover from d to a d-1 dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and T< T(c) is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of T and H . For the Casimir force we have demonstrated that for any sigma > or =1 it is everywhere negative, i.e., an attraction between the surfaces bounding the system is to be observed. At T= T(c) the force is an increasing function of T for sigma>1 and a decreasing one for sigma<1 . For any d and sigma the minimum of the force at T= T(c) is always achieved at some H unequal to 0 .

Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: isotropic case.

We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.

Precise simulation of near-critical fluid coexistence.

We present a novel method to derive liquid-gas coexisting densities, rho(+/-)(T), from grand canonical simulations (without knowledge of T(c) or criticality class). The minima of Q(L) identical with <m(2)>(2)(L)/<m(4)>(L) in an LxLxL box with m=rho-<rho>(L) are used to generate recursively an unbiased universal finite-size scaling function. Monte Carlo data for a hard-core square-well fluid and for the restricted primitive model electrolyte yield rho(+/-) to +/-1%-2% of rho(c) down to 1 part in 10(4)-10(3) of T(c) (and confirm well Ising character). Pressure mixing in the scaling fields is unequivocally revealed and indicates Yang-Yang ratios R(mu)=-0.04(4) and 0.2(6) for the two models, respectively.

Publisher’s Note: Universal finite-size scaling functions with exact nonuniversal metric factors [Phys. Rev. E 67, 065103(R) (2003)

Publisher’s Note: Universal finite-size scaling functions with exact nonuniversal metric factors [Phys. Rev. E 67, 065103(R) (2003)

Universal finite-size scaling functions with exact nonuniversal metric factors

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

On the symmetry of universal finite-size scaling functions in anisotropic systems

In this work a symmetry of universal finite-size scaling functions under a certain anisotropic scale transformation is postulated. This transformation connects the properties of a finite two-dimensional system at criticality with generalized aspect ratio $\rho > 1$ to a system with $\rho < 1$. The symmetry is formulated within a finite-size scaling theory, and expressions for several universal amplitude ratios are derived. The predictions are confirmed within the exactly solvable weakly anisotropic two-dimensional Ising model and are checked within the two-dimensional dipolar in-plane Ising model using Monte Carlo simulations. This model shows a strongly anisotropic phase transition with different correlation length exponents $\nu_{||} \neq \nu_\perp$ parallel and perpendicular to the spin axis.

Universal scaling functions for bond percolation on planar-random and square lattices with multiple percolating clusters.

Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on L1xL2 planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratios L(1)/L(2). We calculate the probability for the appearance of n percolating clusters, W(n); the percolating probabilities P; the average fraction of lattice bonds (sites) in the percolating clusters, <c(b)>(n) (<c(s)>(n)), and the probability distribution function for the fraction c of lattice bonds (sites), in percolating clusters of subgraphs with n percolating clusters, f(n)(c(b)) [f(n)(c(s))]. Using a small number of nonuniversal metric factors, we find that W(n), P, <c(b)>(n) (<c(s)>(n)), and f(n)(c(b)) [f(n)(c(s))] for random lattices, duals of random lattices, and square lattices have the same universal finite-size scaling functions. We also find that nonuniversal metric factors are independent of boundary conditions and aspect ratios.