Kinetic Ising Model Research Articles

Nonequilibrium dynamic transition in a kinetic Ising model driven by both deterministic modulation and correlated stochastic noises

We report the nonequilibrium dynamical phase transition (NDPT) appearing in a kinetic Ising spin system (ISS) subject to the joint application of a deterministic external field and the stochastic mutually correlated noises simultaneously. A time-dependent Ginzburg-Landau stochastic differential equation, including an oscillating modulation and the correlated multiplicative and additive white noises, was addressed and the numerical solution to the relevant Fokker-Planck equation was presented on the basis of an average-period approach of driven field. The correlated white noises and the deterministic modulation induce a kind of dynamic symmetry-breaking order, analogous to the stochastic resonance in trend, in the kinetic ISS, and the reentrant transition has been observed between the dynamic disorder and order phases when the intensities of multiplicative and additive noises were changing. The dependencies of a dynamic order parameterQ upon the intensities of additive noiseA and multiplicative noiseM, the correlation λ between two noises, and the amplitude of applied external fieldh were investigated quantitatively and visualized vividly. Here a brief discussion is given to outline the underlying mechanism of the NDPT in a kinetic ISS driven by an external force and correlated noises.

NONEQUILIBRIUM PHASE TRANSITIONS IN MODEL FERROMAGNETS: A REVIEW

The thermodynamical behaviors of ferromagnetic systems in equilibrium are well studied. However, the ferromagnetic systems far from equilibrium became an interesting field of research in last few decades. Recent exploration of ferromagnetic systems in the presence of a steady magnetic field are also studied by using standard tools of equilibrium statistical physics. The ferromagnet in the presence of time-dependent magnetic field, shows various interesting phenomena. An usual response of a ferromagnet in the presence of a sinusoidally oscillating magnetic field is the hysteresis. Apart from this hysteretic response, the nonequilibrium dynamic phase transition is also a very interesting phenomenon. In this chapter, the nonequilibrium dynamic phase transitions of the model ferromagnetic systems in presence of time-dependent magnetic field are discussed. For this kind of nonequilibrium phase transition, one cannot employ the standard techniques of equilibrium statistical mechanics. The recent developments in this direction are mainly based on numerical simulation (Monte Carlo). The Monte Carlo simulation of kinetic Ising model, in presence of sinusoidally oscillating (in time but uniform over space) magnetic field, is extensively performed to study the nonequilibrium dynamic phase transition. The temperature variations of dynamic order parameter, dynamic specific heat, dynamic relaxation time etc. near the transition point are discussed. The appearance and behaviors of a dynamic length scale and a dynamic time scale near the transition point are also discussed. All these studies indicate that this proposed dynamic transition is a nonequilibrium thermodynamic phase transition. The disorder (quenched) induced zero temperature (athermal) dynamic transition is studied in random field Ising ferromagnet. The dynamic transition in the Heisenberg ferromagnet is also studied. The nature of this transition in the Heisenberg ferromagnet depends on the anisotropy and the polarisation of the applied time varying magnetic field. The anisotropic Heisenberg ferromagnet in the presence of elliptically polarised magnetic field shows multiple dynamic transitions. This multiple dynamic transitions in anisotropic Heisenberg ferromagnet are discussed here. Recent experimental evidences of dynamic transitions are also discussed very briefly.

Phase separation driven by surface diffusion: A Monte Carlo study

We propose a kinetic Ising model to study phase separation driven by surface diffusion. This model is referred to as Model S, and consists of the usual Kawasaki spin-exchange kinetics (Model B) in conjunction with a kinetic constraint. We use multispin coding techniques to develop fast algorithms for Monte Carlo simulations of Models B and S. We use these algorithms to study the late stages of pattern dynamics in these systems, and compare properties of the evolution morphologies, e.g., growth laws, domain distribution functions, and spatial and temporal correlation functions.

Kinetic Ising model in two and three dimensions with constraints

AbstractWe present the dynamics of the kinetically constrained Ising model, comprised of a system of spins coupled with the strength J and situated in a field which plays the role of activation energy. Due to kinetic constraints, glassy effects arise at low temperatures leading to a non‐Arrhenius α‐relaxation time. The results of Monte Carlo simulations are presented for the 2D and 3D facilitated kinetic Ising model with nonzero coupling J. The spin autocorrelation function exhibits, in a broad intermediate time regime, a pronounced stretched exponential decay characterized by an exponent γ. Whereas the relaxation time depends strongly on the activation energy and the interaction strength J, we find a non‐universal stretching exponent γ depending on the temperature and on J. In 3D the autocorrelation function shows a crossover into a pure exponential decay consistent with theoretical predictions. The magnetization is also analyzed and identified as a control parameter of the system. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

First-order reversal curve analysis of homogeneous nucleation in the two-dimensional kinetic Ising model

The first-order reversal curve (FORC) method is applied to the two-dimensional kinetic Ising model. For the system size and magnetic field chosen, the system reverses by the homogeneous nucleation and growth of many droplets. This makes the dynamics of reversal nearly deterministic, in contrast to the strongly disordered systems previously studied by the FORC method. Consequently, the FORC diagrams appear different from those obtained in previous studies. The Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory of phase transformation by nucleation and growth is applied to the system. Reasonable agreement with the Monte Carlo simulations is found, and the FORC method suggests how the KJMA theory could be extended.

Low-temperature dynamics of kinks on Ising interfaces

The anisotropic motion of an interface driven by its intrinsic curvature or by an external field is investigated in the context of the kinetic Ising model in both two and three dimensions. We derive in two dimensions (2D) a continuum evolution equation for the density of kinks by a time-dependent and nonlocal mapping to the asymmetric exclusion process. Whereas kinks execute random walks biased by the external field and pile up vertically on the physical 2D lattice, they execute hard-core biased random walks on a transformed 1D lattice. Their density obeys a nonlinear diffusion equation which can be transformed into the standard expression for the interface velocity, v=M [ (gamma+gamma'') kappa+H] , where M , gamma+gamma", and kappa are the interface mobility, stiffness, and curvature, respectively. In 3D, we obtain the velocity of a curved interface near the 100 orientation from an analysis of the self-similar evolution of 2D shrinking terraces. We show that this velocity is consistent with the one predicted from the 3D tensorial generalization of the law for anisotropic curvature-driven motion. In this generalization, both the interface stiffness tensor and the curvature tensor are singular at the 100 orientation. However, their product, which determines the interface velocity, is smooth. In addition, we illustrate how this kink-based kinetic description provides a useful framework for studying more complex situations by modeling the effect of immobile dilute impurities.

Dynamical real-space renormalization group calculations with a highly connected clustering scheme on disordered networks

We have defined a type of clustering scheme preserving the connectivity of the nodes in a network, ignored by the conventional Migdal-Kadanoff bond moving process. In high dimensions, our clustering scheme performs better for correlation length and dynamical critical exponents than the conventional Migdal-Kadanoff bond moving scheme. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising model to be z=2.13 and z=2.09 , respectively, at the pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behavior of randomly bond diluted lattices in d=2 and 3 in the light of this transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law and Poissonian degree distributions.

Entropic sampling dynamics of the globally coupled kinetic Ising model

The entropic sampling dynamics based on the reversible information transfer to and from the environment is applied to the globally coupled Ising model in the presence of an oscillating magnetic field. When the driving frequency is low enough, coherence between the magnetization and the external magnetic field is observed; such behavior tends to weaken with the system size. The time-scale matching between the intrinsic time scale, defined in the absence of the external magnetic field, and the extrinsic time scale, given by the inverse of the driving frequency, is used to explain the observed coherence behavior.

Nonequilibrium dynamic transition in a kinetic Ising model driven by both deterministic modulation and correlated stochastic noises

We report the nonequilibrium dynamical phase transition (NDPT) appearing in a kinetic Ising spin system (ISS) subject to the joint application of a deterministic external field and the stochastic mutually correlated noises simultaneously. A time-dependent Ginzburg-Landau stochastic differential equation, including an oscillating modulation and the correlated multiplicative and additive white noises, was addressed and the numerical solution to the relevant Fokker-Planck equation was presented on the basis of an average-period approach of driven field. The correlated white noises and the deterministic modulation induce a kind of dynamic symmetry-breaking order, analogous to the stochastic resonance in trend, in the kinetic ISS, and the reentrant transition has been observed between the dynamic disorder and order phases when the intensities of multiplicative and additive noises were changing. The dependencies of a dynamic order parameter Q upon the intensities of additive noise A and multiplicative noise M, the correlation lmda between two noises, and the amplitude of applied external field h were investigated quantitatively and visualized vividly. A brief discussion was given to outline the underlying mechanism of the NDPT in a kinetic ISS driven by an external force and correlated noises. Keywords: Ising spin system, nonequilibrium dynamical phase transition, stochastic resonance, correlated noises, TDGL model. PACS: 75.10.Hk, 64.60.Ht, 05.10.Gg, 76.20.+q

Importance of proper choice of transition rates in kinetic simulations of dynamic processes

Proper transition rates in kinetic mean-field and Monte Carlo simulations of dynamic processes are very important to obtain realistic results. We show that by proper choice of the transition rates one can unify the advantages and eliminate the disadvantages of different models used in the literature [see, e.g., Senhaji et al., Surf. Sci. 274, 297 (1992 )( kinetic tight binding Ising model), Martin, Phys. Rev. B 41, 2279 (1990) and Cserhati et al., Surf. Sci. 290, 345 (1993)]. Furthermore, we also show that this choice cannot be considered simply as an extension of the previous ones. It contains an additional parameter controlling the ratio of the transition rates in the bulk and close to the surface sGsurf / Gbulkd, which influences the kinetics of surface segregation. We show that some results obtained previously need reconsideration. We illustrate, e.g., how the composition dependence of the transition rates (diffusion coefficient) influences the ‘surfactant formation and dissolution mode’. For example, the dissolution kinetics can deviate from the parabolic law on the nanoscale in accordance with our recent results [Erdelyi et al., Phys. Rev. B 69, 113407 (2004)]. Furthermore, we also present how the sGsurf / Gbulkd ratio affects the kinetic segregation isotherm.

Length Dependent Helix-Coil Transition Kinetics of Nine Alanine-Based Peptides.

It is well-known that end caps and the peptide length can dramatically influence the thermodynamics of the helix-coil transition. However, their roles in determining the kinetics of the helix-coil transition have not been studied extensively and are less well understood. Kinetic Ising models and sequential kinetic models involving barrier crossing via diffusion all predict that the helix formation time depends monotonically on the peptide length with the relaxation time increasing with respect to increasing chain length. Here, we have studied the helix-coil transition kinetics of a series of Ala-based α-helical peptides of different length (19-39 residues), with and without end caps, using time-resolved infrared spectroscopy coupled with laser-induced temperature jump (T-jump) initiation method. The helical content of these peptides was kinetically monitored by probing the amide carbonyl stretching frequencies (i.e., the amide I' band) of the peptide backbone. We found that the relaxation rates for peptides with efficient end caps are more rapid than those of the corresponding peptides without good end caps. These results indicate that efficient end-capping sequences can not only stabilize preexisting helices but also promote helix formation through initiation. Furthermore, we found that the relaxation times of these peptides, following a T-jump of 1-11 °C, show rather complex behaviors as a function of the peptide length, in disagreement with theoretical predications. Theses results are not readily explained by theories in which Ala is taken to have a single helical propensity (s). However, recent studies have suggested that s depends on chain length; when this factor is considered, the mean first-passage times of the coil-to-helix transition show similar dependence on the peptide length as those observed experimentally.

Low-temperature nucleation in a kinetic Ising model under different stochastic dynamics with local energy barriers.

Using both analytical and simulational methods, we study low-temperature nucleation rates in kinetic Ising lattice-gas models that evolve under two different Arrhenius dynamics that interpose between the Ising states a transition state representing a local energy barrier. The two dynamics are the transition-state approximation [T. Ala-Nissila, J. Kjoll, and S. C. Ying, Phys. Rev. B 46, 846 (1992)] and the one-step dynamic [H. C. Kang and W. H. Weinberg, J. Chem. Phys. 90, 2824 (1989)]. Even though they both obey detailed balance and are here applied to a situation that does not conserve the order parameter, we find significant differences between the nucleation rates observed with the two dynamics, and between them and the standard Glauber dynamic [R. J. Glauber, J. Math. Phys. 4, 294 (1963)], which does not contain transition states. Our results show that great care must be exercised when devising kinetic Monte Carlo transition rates for specific physical or chemical systems.

Is a "homogeneous" description of dynamic heterogeneities possible?

We study the simplest model of dynamic heterogeneities in glass forming liquids: one-spin facilitated kinetic Ising model introduced by Fredrickson and Andersen [G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. 53, 1244 (1984); J. Chem. Phys. 83, 5822 (1985)]. We show that the low-temperature, long-time behavior of the density autocorrelation function predicted by a scaling approach can be obtained from a self-consistent mode-couplinglike approximation.

Scaling and universality of inherent structure simulations.

In this paper we explore the inherent structures (IS) approach to the dynamics of the East constrained kinetic Ising model. The inherent structures do not capture the nature of the dynamics of many quantities, including the spin autocorrelation function. Simply monitoring the quenched energy fluctuations, i.e., IS energy, results in an oversimplified single order-parameter description of the system's dynamics, but examining other features, such as domain dynamics or normal modes, may give a more complete and useful picture of the dynamics. The universality in the behavior of the IS energy of this model does not reveal nonuniversal features of the kinetics that determine long-time relaxation of the system. As a result, popular functional forms, such as the stretched exponential relaxation or Gaussian distribution of energies, may be a numerical fit to data with little physical justification. Filtering data can be shown to erase features of the system and the resulting quantities resemble more universal functional forms that lack physical insight. These results for the East model have implications for IS simulations of realistic systems and suggest careful analysis including the examination of other potential order parameters is necessary to evaluate the validity of applications of universal and scaling arguments to IS simulations.

AGEING, DYNAMICAL SCALING AND CONFORMAL INVARIANCE

Building on an analogy with conformal invariance, local scale transformations consistent with dynamical scaling are constructed. Two types of local scale invariance are found which act as dynamical space–time symmetries of certain nonlocal free field theories. The scaling form of two-point functions is completely fixed by the requirement of local scale invariance. These predictions are confirmed through tests in the 3D ANNNI model at its Lifshitz point and in ageing phenomena of simple ferromagnets, here studied through the kinetic Ising model with Glauber dynamics.

Using Kinetic Monte Carlo simulations to study phase separation in Alloys

We review recent extensions of the kinetic Ising model used to investigate phase separation in binary alloys. Firstly, vacancies are included to model the diffusion of the atoms on the microscopic scale more realistically. These can change the coarsening rate and the coarsening mechanism. Secondly, the lattice is allowed to deform owing to the different sizes of the atoms and the resulting misfit between precipitates and matrix. The deformability of the lattice induces long-range elastic interactions between the atoms. These change the shape, orientation, and arrangement of the precipitates. The growth of the precipitates need not follow the law.

Linear time dependence of the surfactant effect: A local equilibrium under flux

We have investigated the dissolution kinetics of an A metal thin film deposited on a (100) B metal under thermal annealing for a system with a strong B segregation (e.g., Ni/Ag). The interdiffusion is described using both Monte Carlo and mean-field simulations with an effective exchange mechanism derived from the kinetic tight-binding Ising model. Surprisingly, the surface enrichment in substrate B element is found to increase linearly with time instead of following the usual $\sqrt{t}$ law. We demonstrate that this t law occurs when the diffusion coefficient of B atoms in the A deposit is high or when the deposit thickness decreases. Moreover it is shown that, in this surfactant regime, the usual local equilibrium rule which drives the kinetics has to be generalized under the flux of B atoms coming from the deposit/substrate interface. Consequences of this new formulation on nonequilibrium dynamical behaviors during thin film dissolutions are discussed.

Phase transformation in a lattice system in the presence of spin-exchange dynamics

A joint action of the Glauber single-spin-flip and the Kawasaki spin-exchange mechanisms upon the processes of phase transformation is examined in the framework of the one-dimensional kinetic Ising model. It is shown that the addition of the Kawasaki dynamics to that of Glauber accelerates the process of phase transformation in the initial stage, but slows it down in later stages. For the truncated form of Glauber dynamics, which excludes the processes of splitting and coagulation of clusters, the addition of the Kawasaki dynamics always accelerates the phase transformation process. Acting alone, the Kawasaki mechanism provides a cluster growth proportional to t(1/2) (where t is the time) in the initial stage and proportional to t(1/3) (Lifshitz-Slyozov-Wagner law) in the intermediate stage. In the final stage, a cluster size approaches exponentially its equilibrium value.

Low-temperature nucleation in a kinetic Ising model with soft stochastic dynamics.

We study low-temperature nucleation in kinetic Ising models by analytical and simulational methods, confirming the general result for the average metastable lifetime, <tau>=Aexp((betagamma) (beta=1/k(B)T) [Commun. Math. Phys. 137, 209 (1991)]]. Contrary to common belief, we find that both A and Gamma depend significantly on the stochastic dynamic. In particular, for a "soft" dynamic, in which the effects of the interactions and the applied field factorize in the transition rates, Gamma does not simply equal the energy barrier against nucleation, as it does for the standard Glauber dynamic, which does not have this factorization property.

On the universality of the fluctuation–dissipation ratio in non-equilibrium critical dynamics

The two-time non-equilibrium correlation and response functions in 1D kinetic classical spin systems with non-conserved dynamics and quenched to their zero-temperature critical point are studied. The exact solution of the kinetic Ising model with Glauber dynamics for a wide class of initial states allows for an explicit test of the universality of the non-equilibrium limit fluctuation–dissipation ratio X∞. It is shown that the value of X∞ depends on whether the initial state has finitely many domain walls or not and thus two distinct dynamic universality classes can be identified in this model. Generic 1D kinetic spin systems with non-conserved dynamics fall into the same universality classes as the kinetic Glauber–Ising model provided the dynamics is invariant under the C-symmetry of simultaneous spin and magnetic-field reversal. While C-symmetry is satisfied for magnetic systems, it need not be for lattice gases which may therefore display hitherto unexplored types of non-universal kinetics.