Kinetic Ising Model Research Articles

Mixed-spin ising model with one- and two-spin competing dynamics

In this work we found the stationary states of a kinetic Ising model, with two different types of spins: sigma=1/2 and S=1. We divided the spins into two interpenetrating sublattices, and found the time evolution for the probability of the states of the system. We employed two transition rates which compete between themselves: one, associated with the Glauber process, which describes the relaxation of the system through one-spin flips; the other, related to the simultaneous flipping of pairs of neighboring spins, simulates an input of energy into the system. Using the dynamical pair approximation, we determined the equations of motion for the sublattice magnetizations, and also for the correlation function between first neighbors. We found the phase diagram for the stationary states of the model, and we showed that it exhibits two continuous transition lines: one line between the ferrimagnetic and paramagnetic phases, and the other between the paramagnetic and antiferrimagnetic phases.

Analytic Approximations for the Velocity of Field-Driven Ising Interfaces

We present analytic approximations for the field, temperature, and orientation dependences of the interface velocity in a two-dimensional kinetic Ising model in a nonzero field. The model, which has nonconserved order parameter, is useful for ferromagnets, ferroelectrics, and other systems undergoing order–disorder phase transformations driven by a bulk free-energy difference. The solid-on-solid (SOS) approximation for the microscopic surface structure is used to estimate mean spin-class populations, from which the mean interface velocity can be obtained for any specific single-spin-flip dynamic. This linear-response approximation remains accurate for higher temperatures than the single-step and polynuclear growth models, while it reduces to these in the appropriate low-temperature limits. The equilibrium SOS approximation is generalized by mean-field arguments to obtain field-dependent spin-class populations for moving interfaces, and thereby a nonlinear-response approximation for the velocity. The analytic results for the interface velocity and the spin-class populations are compared with Monte Carlo simulations. Excellent agreement is found in a wide range of field, temperature, and interface orientation.

Evidence of a dynamical length scale in the `f-spin frustrated kinetic Ising model'

In this paper we study the connection between the equilibrium relaxation properties of glass-forming systems and the spatial organization of cooperative processes which govern their dynamics. This study has been performed by Monte Carlo simulations of f-spin frustrated kinetic lsing models for different local kinetic constraints f. Both the equilibrium dynamics of these systems and the topological properties of their cooperative processes were investigated through the calculation of autocorrelation functions and the distribution of cooperativity lengths. The results show that the different local kinetic constraints induce very different spatial organizations of the cooperative processes and thus heterogeneous dynamics of different natures. On the other hand, whatever their spatial organization, the average spatial extent of the cooperative processes acts as a dynamical length scale which governs the dynamics of the system.

Layer-by-layer versus surfactant dissolution modes in heteroepitaxy

~Received 18 May 1999! Dissolution of a thick A film into aB substrate during metal heteroepitaxy ( A/B) obeys general trends which are described here in the case where the corresponding AB alloy presents a tendency to bulk phase separation and to B surface segregation ~e.g., A5Ni, B5Ag). Using the kinetic tight-binding Ising model, either in the mean-field or in the Monte Carlo framework, we findlayer-by-layer dissolution modes which, depending on the annealing temperature T, are preceded ( T.Tk) or not ( T,Tk) by the rising of capping B monolayers burying an almost intact A film ~surfactantlike effect!. Tk is found to decrease as the tendency to surface segregation of the substrate element increases, which can be understood in terms of local equilibrium at both the surface and the interface of the deposit. The model gives access to the main kinetic laws which are obeyed in the two dissolution modes, i.e., the total quantity of deposit matter decreases linearly with the square root of time. @S0163-1829~99!00243-X#

Facilitated Ising Model with Soft Restrictions

An analytical approach is proposed to include dynamical constraints into a kinetic Ising model where the spins are subjected to flip processes. The relaxation time shows a non-Arrhenius behaviour. When the constraints are realized with a certain probability the relaxation time decreases further. For that case a general relation is derived the validity of which is demonstrated for different probability distributions. As a special case we consider a deterministic constraint regarded as a voter model, which may lead to an unstable behaviour.

Crossover from multiscaling to standard scaling in systems with conserved scalar order parameter

The phase-ordering dynamics with conserved scalar order parameter is studied simulating the Cahn–Hilliard equation and the kinetic Ising model. In both cases a preasymptotic multiscaling regime is found revealing that the late stage of phase-ordering is always approached through a crossover from multiscaling to standard scaling, independently from the nature of the microscopic dynamics.

Monte Carlo simulation of solute aggregation in binary alloys: Domain boundary precipitation and domain growth

Domain boundary precipitation and domain growth in binary polycrystalline materials are studied by applying the Monte Carlo simulation in two-dimensional squared lattice. The simulation is carried out on a hybrid lattice of the kinetic spin-exchange Ising model coupled with the Q-state Potts model. First of all, the static properties of this coupled model are studied, predicting just a small shiftdown of the critical point with enhanced Potts interactions. Subsequently, the dynamic properties, such as morphology and coarsening kinetics of the second phase precipitates as well as kinetics and scaling of the domain growth, are investigated in detail. Pronounced second phase precipitation at domain boundaries is observed at a temperature range just below critical point ${T}_{c}$ as long as the solutes prefer to segregate onto the boundaries. However, the boundary precipitation is significantly prohibited at either high or low temperatures $(T\ensuremath{\gg}{T}_{c}$ or $T\ensuremath{\ll}{T}_{c}).$ We demonstrate that the domain growth is slowed down due to the pinning effect of the precipitates at the boundaries, no matter what the boundary migration ability is. The kinetics of boundary precipitation and domain growth in various systems are simulated. Both the Lifshitz-Slyozov-Wagner law for second phase coarsening and the linear law for the normal domain growth become broken due to the domain boundary precipitation. The scaling behavior of the domain growth is identified in present systems although a further confirmation may be required.

Triangular anisotropies in driven diffusive systems: reconciliation of up and down.

Deterministic coarse-grained descriptions of driven diffusive systems (DDS) have been hampered by apparent inconsistencies with the kinetic Ising models of DDS. In the evolution towards the driven steady-state, "triangular" anisotropies in the two systems point in opposite directions with respect to the drive field. We show that this is nonuniversal behavior in the sense that the triangular anisotropy "flips" with local modifications of the Ising interactions. The sign and magnitude of the triangular anisotropy also vary with temperature. We have also flipped the anisotropy of coarse-grained models, though not yet at the latest stages of evolution. Our results illustrate the comparison of deterministic coarse-grained and stochastic Ising DDS studies to identify universal phenomena in driven systems. Coarse-grained systems are particularly attractive in terms of analysis and computational efficiency.

Parallelization of a Dynamic Monte Carlo Algorithm: A Partially Rejection-Free Conservative Approach

We experiment with a massively parallel implementation of an algorithm for simulating the dynamics of metastable decay in kinetic Ising models. The parallel scheme is directly applicable to a wide range of stochastic cellular automata where the discrete events (updates) are Poisson arrivals. For high performance, we utilize a continuous-time, asynchronous parallel version of the n-fold way rejection-free algorithm. Each processing element carries an l×l block of spins, and we employ fast one-sided communication routines on a distributed-memory parallel architecture. Different processing elements have different local simulated times. To ensure causality, the algorithm handles the asynchrony in a conservative fashion. Despite relatively low utilization and an intricate relationship between the average time increment and the size of the spin blocks, we find that the algorithm is scalable and for sufficiently large l it outperforms its corresponding parallel Metropolis (non-rejection-free) counterpart. As a sample application, we present results for metastable decay in a model ferromagnetic or ferroelectric film, observed with a probe of area smaller than the total system.

AN ANALYTICAL APPROACH TO THE FREDRICKSON–ANDERSEN MODEL WITH VACANCIES

The dynamics of the n-spin facilitated kinetic Ising model (Fredrickson–Andersen model) with mobile vacancies as a model for the glassy materials are studied analytically by means of the Fock-space representation of the master equation. The system is mapped onto a three state model characterizing mobile, immobile and vacant cells. The characteristic cooperativity for glass forming systems are introduced by restrictions influencing the local dynamics and subsequently the local mobility of different lattice cells. In a moderate temperature regime the relaxation time versus the inverse temperature T-1 reveals two processes. Whereas the slow process can be identified with the conventional α-process of the supercooled liquid, the fast one originated by the additional empty sites is suggested to be the β JG -process due to Johari–Goldstein. The results are accordant with numerical simulations and suggest that the modified n-spin facilitated kinetic Ising model is able to describe qualitatively the behaviour of a supercooled liquid near the glass transition temperature T g .

Nontrivial stochastic resonance temperature for the kinetic Ising model

The kinetic Ising model in a weak oscillating magnetic field is studied in the context of stochastic resonance. The signal-to-noise ratio calculated with simulations is found to peak at a nontrivial resonance temperature above the equilibrium critical temperature ${T}_{c}.$ We argue that its appearance is closely related to the vanishing of the kinetic coefficient at ${T}_{c}.$ Comparisons with various theoretical results in one and higher dimensions are made.

Kinetic Ising model in an oscillating field: Avrami theory for the hysteretic response and finite-size scaling for the dynamic phase transition

Hysteresis is studied for a two-dimensional, spin-1/2, nearest-neighbor, kinetic Ising ferromagnet in an oscillating field, using Monte Carlo simulations and analytical theory. Attention is focused on large systems and strong field amplitudes at a temperature below T_c. In this parameter regime, the magnetization switches through random nucleation and subsequent growth of many droplets of spins aligned with the applied field. Using a time-dependent extension of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of metastable decay, we analyze the statistical properties of the hysteresis-loop area and the correlation between the magnetization and the applied field. This analysis enables us to accurately predict the results of extensive Monte Carlo simulations. The average loop area exhibits an extremely slow approach to an asymptotic, logarithmic dependence on the product of the amplitude and the frequency of the applied field. This may explain the inconsistent exponent estimates reported in previous attempts to fit experimental and numerical data for the low-frequency behavior of this quantity to a power law. At higher frequencies we observe a dynamic phase transition. Applying standard finite-size scaling techniques from the theory of second-order equilibrium phase transitions to this nonequilibrium phase transition, we obtain estimates for the transition frequency and the critical exponents (beta/nu \approx 0.11, gamma/nu \approx 1.84 and nu \approx 1.1). In addition to their significance for the interpretation of recent experiments on switching in ferromagnetic and ferroelectric nanoparticles and thin films, our results provide evidence for the relevance of universality and finite-size scaling to dynamic phase transitions in spatially extended nonstationary systems.

Monte Carlo simulations of a generalizedn-spin facilitated kinetic Ising model

A kinetic Ising model is analyzed where spin variables correspond to lattice cells with mobile or immobile particles. Introducing additional restrictions for the flip processes according to the n spin-facilitated kinetic Ising model and using Monte Carlo methods, we study the freezing process under the influence of an additional nearest-neighbor interaction. The stretched exponential decay of the auto correlation function is observed and the exponent $\ensuremath{\gamma},$ as well as the relaxation time are determined depending on the activation energy h and the short-range coupling J. The magnetization corresponding to the density of immobile particles is found to be the controlling parameter for the dynamic evolution.

Theoretical prediction of new dissolution modes during metal heteroepitaxy

By means of the kinetic tight-binding Ising model we investigate the Ni/Ag dissolution modes resulting from the thermal interdiffusion between the two elements. For a thin deposit (1 ML) we show that the dissolution proceeds by the formation of encapsulated Ni precipitates covered by two pure Ag monolayers. For a thick deposit (10 ML) we find two distinct layer-by-layer modes depending on the annealing temperature. At high temperature the kinetics start with the apparition of capping Ag monolayers on a buried almost intact Ni film ( surfactant-layer-by-layer mode). Below a critical temperature T c, the dissolution proceeds without any Ag surface enrichment ( layer-by-layer mode). We find that T c decreases when the surface driving forces which favour the segregation of the substrate elements increase.

Thermodynamics of an extended Fredrickson-Andersen model

We describe an extension of the Fredrickson-Andersen model (or n-spin facilitated kinetic Ising model) which exhibits a transition similar to a glass transition. Our extension incorporates long-range effects. Exact or numerically accurate results are obtained only for one dimension, but most conclusions apply more generally. The model exhibits a discontinuous drop in the specific heat as it is cooled through the transition temperature. This leads to an excess internal energy and a residual entropy at zero temperature. The disorder associated with the residual entropy can be seen in diffuse scattering which is characteristic of disordered systems.

Dynamic Response of an Ising System to a Time-Dependent Field

The basic equation of a kinetic Ising model under the influence of a time dependent field is rederived applying the Fock space representation of the master equation. The nonequilibrium behaviour of the model in the presence of both a static and an oscillating magnetic field is analyzed by numerical simulations and by scaling arguments. Forcing the system to be initially in the metastable state and imposing an additional oscillating field makes it possible to study the relaxation time of this state. We find a power law behaviour for the waiting time depending on the amplitude and the frequency of the field as well as on the reduced temperature. The critical exponents are determined numerically. Furthermore, the results are supported by scaling arguments.

Nonequilibrium phase transition in the kinetic Ising model: Existence of a tricritical point and stochastic resonance

The dynamic phase transition has been studied in the two dimensional kinetic Ising model in presence of a time varying (sinusoidal) magnetic field by Monte Carlo simulation. The nature (continuous or discontinuous) of the transition is characterized by studying the distribution of the order parameter and the temperature variation of the fourth order cumulant. For the higher values of the field amplitude, the transition observed is discontinuous and it is continuous for lower values of the field amplitude, indicating the existence of a tricritical point (separating the nature of the transition) on the phase boundary. The transition is observed to be a manifestation of stochastic resonance.

Feedback coupling and self-organization

The correlation function of a kinetic Ising model offers a crossover from an exponential to an algebraic decay with a universal exponent when a feedback coupling is introduced via a stochastic field. The correlation of this field is directly related to the current spin correlation function. Due to the feedback coupling, at low temperatures there appears a strong slowing down resulting in a self-similar behaviour. We argue that such a self-organized mechanism should be relevant in glasses.

A Monte–Carlo approach to domain boundary precipitation in binary alloys

A Monte–Carlo approach on the second phase precipitation in polycrystalline binary alloys is developed. The approach starts from the kinetic spin-exchange Ising model and the Q-state Potts model, and a coupling algorithm for simulation of both phase precipitation and domain growth is developed. The simulation on a simplified system with slow domain boundary migration reveals precipitation phenomena on the domain boundaries. The effect of domain boundaries on the morphology and kinetics of the second phase precipitates is investigated. It is shown that the scaling concept and the Lifshitz–Slyozov–Wagner law are broken in the present system.

Stationary definition of persistence for finite-temperature phase ordering

For the two dimensional kinetic Ising model at finite temperature, the local mean magnetisation $M_t=t^{-1}\int_{0}^t\sigma(t')\d t'$, simply related to the fraction of time spent by a given spin in the positive direction, has a limiting distribution, singular at $\pm m_0(T)$, the Onsager spontaneous magnetization. The exponent of this singularity defines the persistence exponent $\theta$. We also study first passage exponents associated to persistent large deviations of $M_t$, and their temperature dependence.