We analyse the spin-flip dynamics in kinetic Ising chains with Kimball-Deker-Haake (KDH) transition rates, and evaluate exactly the evolution of global quantities like magnetisation and its fluctuations, and the two-time susceptibilities and correlations of the global spin and the global three-spin. Information on the ageing behaviour after a quench to zero temperature is extracted.
The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform disorder is investigated by intensive Monte Carlo simulations. Taking into account finite-time corrections to scaling, simple ageing behaviour is observed in the two-time responses and correlators. The dynamical exponent z and the form of the scaling functions only depend on the ratio eps/T, where eps describes the width of the distribution of the disorder. The agreement of the predictions of local scale-invariance generalised to z<> 2 for the two-time scaling functions of response and correlations with t…
The local scale invariance has been investigated in the nonequilibrium kinetic Ising model exhibiting absorbing phase transition of PC type in 1+1 dimension. Numerical evidence has been found for the satisfaction of this symmetry and estimates for the critical ageing exponents are given.
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.
We show that a nominal temperature can be consistently and uniquely defined everywhere in the phase diagram of large classes of nonequilibrium kinetic Ising spin models. In addition, we confirm the recent proposal that, at critical points, the large-time ``fluctuation-dissipation ratio'' $X_\infty$ is a universal amplitude ratio and find in particular $X_\infty \approx 0.33(2)$ and $X_\infty = 1/2$ for the magnetization in, respectively, the two-dimensional Ising and voter universality classes.
The exact energy spectrum is developed for a two temperature kinetic Ising spin chain, and its dual reaction diffusion system with spatially alternating pair annihilation and creation rates. Symmetries of the system pseudo-Hamiltonian that enable calculation of the spectrum are also used to derive explicit state vectors for small system sizes, and to make observations regarding state vectors in the general case. Physical consequences of the surprisingly simple form for the eigenvalues are also discussed.
The evolution of the structure factor is studied during the phase-ordering dynamics of the kinetic Ising model with conserved order parameter. A preasymptotic multiscaling regime is found as in the solution of the Cahn-Hilliard-Cook equation, 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.
Recently Baumann et al. [arXiv:0709.3228v1] studied the phase-ordering kinetics of the two-dimensional Ising model with uniform spatially quenched disorder by Monte-Carlo simulations. They found that the two-time response and correlation functions are in agreement with the predictions of local scale invariance generalised to z!=2. The present paper shows why this is not true and suggests an alternative approach which leads to a much better agreement with the numerical results.
Based on dynamical cavity method, we propose an approach to the inference of kinetic Ising model, which asks to reconstruct couplings and external fields from given time-dependent output of original system. Our approach gives an exact result on tree graphs and a good approximation on sparse graphs, it can be seen as an extension of Belief Propagation inference of static Ising model to kinetic Ising model. While existing mean field methods to the kinetic Ising inference e.g., na\" ive mean-field, TAP equation and simply mean-field, use approximations which calculate magnetizations and correlati…
We introduce a quantum generalization of classical kinetic Ising models, described by a certain class of quantum many body master equations. Similarly to kinetic Ising models with detailed balance that are equivalent to certain Hamiltonian systems, our models reduce to a set of Hamiltonian systems determining the dynamics of the elements of the many body density matrix. The ground states of these Hamiltonians are well described by matrix product, or pair entangled projected states. We discuss critical properties of such Hamiltonians, as well as entanglement properties of their low energy state…
Using numerical simulations we investigate the properties of the dynamic phase transition that is encountered in the three-dimensional Ising model subjected to a periodically oscillating magnetic field. The values of the critical exponents are determined through finite-size scaling. Our results show that the studied non-equilibrium phase transition belongs to the universality class of the equilibrium three-dimensional Ising model.
The d-dimensional n-spin facilitated kinetic Ising model is studied analytically starting from usual master equations and their transformation into a Fock-space representation. The evolution of relevant operators is rewritten in terms of a projection formalism. The obtained frequency matrices and memory terms are analyzed. Especially, the influences of the memory terms is approached by using standard techniques of the usual mode coupling approach. The temperature dependence of the relaxation times related to the n-spin facilitated kinetic Ising model shows a weak non-Arrhenius behavior. Furthe…
Ising models obeying Glauber dynamics in a temporally oscillating magnetic field are analyzed. In the context of stochastic resonance, the response in the magnetization is calculated by means of both a mean-field theory with linear-response approximation, and the time-dependent Ginzburg-Landau equation. Analytic results for the temperature and frequency dependent response, including the resonance temperature, compare favorably with simulation data.
A kinetic Ising model description of Liesegang phenomena is studied using Monte Carlo simulations. The model takes into account thermal fluctuations, contains noise in the chemical reactions, and its control parameters are experimentally accessible. We find that noisy, irregular precipitation takes place in dimension d=2 while, depending on the values of the control parameters, either irregular patterns or precipitation bands satisfying the regular spacing law emerge in d=3.
We investigate how the time evolution of different kinetic Ising models depends on the initial conditions of the dynamics. To this end we consider the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the time evolution of a joint probability distribution of the two systems. This equation is then solved within an effective-field approach. By analyzing the fixed points of the master equation and their stability we identify regular and chaotic phases.
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 $γ$ as well as the relaxation time are determined depending on the activation energy $h$ and the short range coupling $J$. The magnetization corre…
We show using scaling arguments and Monte Carlo simulations that a class of binary interacting models of opinion evolution belong to the Ising universality class in presence of an annealed noise term of finite amplitude. While the zero noise limit is known to show an active-absorbing transition, addition of annealed noise induces a continuous order-disorder transition with Ising universality class in the infinite-range (mean field) limit of the models.
We consider a model for thermal contact through a diathermal interface between two macroscopic bodies at different temperatures: an Ising spin chain with nearest neighbor interactions is endowed with a Glauber dynamics with different temperatures and kinetic parameters on alternating sites. The inhomogeneity of the kinetic parameter is a novelty with respect to the model of Ref.[1] and we exhibit its influence upon the stationary non equilibrium values of the two-spin correlations at any distance. By mapping to the dynamics of spin domain walls and using free fermion techniques, we determine t…
We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in differen…
We present a comparative study of the fate of an Ising ferromagnet on the square lattice with periodic boundary conditions evolving under three different zero-temperature dynamics. The first one is Glauber dynamics, the two other dynamics correspond to two limits of the directed Ising model, defined by rules that break the full symmetry of the former, yet sharing the same Boltzmann-Gibbs distribution at stationarity. In one of these limits the directed Ising model is reversible, in the other one it is irreversible. For the kinetic Ising-Glauber model, several recent studies have demonstrated t…
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are investigated here numerically from the point of view of the underlying spin system. The dynamical persistency exponent $Θ$ and the exponent $lambda$ characterising the two-time autocorrelation function of the total magnetization under non-equilibrium conditions are reported. It is found that the PC transition has strong effect: the process becomes non-Markovi…
The effects of locally broken spin symmetry are investigated in one dimensional nonequilibrium kinetic Ising systems via computer simulations and cluster mean field calculations. Besides a line of directed percolation transitions, a line of transitions belonging to N-component, two-offspring branching annihilating random-walk class (N-BARW2) is revealed in the phase diagram at zero branching rate. In this way a spin model for N-BARW2 transitions is proposed for the first time.
We study the phase-ordering kinetics following a quench to a final temperature $T_f$ of the one-dimensional p-state clock model. We show the existence of a critical value $p_c=4$, where the properties of the dynamics change. At $T_f=0$, for $p\le p_c$ the dynamics is analogous to that of the kinetic Ising model, characterized by Brownian motion and annihilation of interfaces. Dynamical scaling is obeyed with the same dynamical exponents and scaling functions of the Ising model. For $p>p_c$, instead, the dynamics is dominated by a texture mechanism analogous to the one-dimensional XY model, and…
We use an efficient method that eases the daunting task of simulating dynamics in spin systems with long-range interaction. Our Monte Carlo simulations of the long-range Ising model for the nonequilibrium phase ordering dynamics in two spatial dimensions erform significantly faster than the standard Metropolis approach and considerably more efficiently than the kinetic Monte Carlo method. Importantly, this enables us to establish agreement with the theoretical prediction for the time dependence of domain growth, in contrast to previous numerical studies. This method can easily be generalized t…
If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the…
The exact relation between the response function $R(t,t^{\prime})$ and the two time correlation function $C(t,t^{\prime})$ is derived analytically in the one dimensional kinetic Ising model subjected to a temperature quench. The fluctuation dissipation ratio $X(t,t^{\prime})$ is found to depend on time through $C(t,t^{\prime})$ in the time region where scaling $C(t,t^{\prime}) = f(t/t^{\prime})$ holds. The crossover from the nontrivial form $X(C(t,t^{\prime}))$ to $X(t,t^{\prime}) \equiv 1$ takes place as the waiting time $t_w$ is increased from below to above the equilibration time $t_{eq}$.
The dynamic critical exponent and the frequency and wave-vector dependent susceptibility of the kinetic Ising model with Glauber dynamics on an alternating isotopic chain are examined. The analysis provides to our knowledge the first connection between a microscopic model and the Nagel scaling curve originally proposed to describe dielectric susceptibility measurements of several glass-forming liquids. While support is given to the hypothesis relating the Nagel scaling to multiple relaxation processes, it is also found that the scaling function may also exhibit plateau regions and does not hol…
The nonequilibrium dynamic phase transition, in the two dimensional kinetic Ising model in presence of a randomly varying (in time but uniform in space) magnetic field, has been studied both by Monte Carlo simulation and by solving the mean field dynamic equation of motion for the average magnetisation. In both the cases, the time averaged magnetisation vanishes from a nonzero value depending upon the values of the width of randomly varying field and the temperature. The phase boundary lines are drawn in the plane formed by the width of the random field and the temperature.
We study hysteresis for a two-dimensional, spin-1/2, nearest-neighbor, kinetic Ising ferromagnet in an oscillating field, using Monte Carlo simulations. The period-averaged magnetization is the order parameter for a proposed dynamic phase transition (DPT). To quantify the nature of this transition, we present the first finite-size scaling study of the DPT for this model. Evidence of a diverging correlation length is given, and we provide estimates of the transition frequency and the critical indices $β$, $γ$ and $ν$.
In this paper we study generalised Ising Glauber models with inflow of informa- tion in one dimension and derive expressions for the exit probability using well established analytical methods. The analytical expressions agree very well with the results obtained from numerical simulation only when the interaction is restricted to the nearest neighbor. But as the range of interaction is increased the analytical results deviate from simulation results systematically. The reasons for the deviation as well as some related open questions are discussed.
The nonequilibrium dynamic phase transition, in the two dimensional site diluted kinetic Ising model in presence of an oscillating magnetic field, has been studied by Monte Carlo simulation. The projections of dynamical phase surface are drawn in the planes formed by the dilution and field amplitude and the plane formed by temperature and field amplitude. The tricritical behaviour is found to be absent in this case which was observed in the pure system.
A nonuniform extension of the Glauber model on a one-dimensional lattice with boundaries is investigated. Based on detailed balance, reaction rates are proposed for the system. The static behavior of the system is investigated. It is shown that there are cases where the system exhibits a static phase transition, which is a change of behavior of the static profile of the expectation values of the spins near end points.
We investigate the long-time properties of the Ising-Glauber model on a periodic cubic lattice after a quench to zero temperature. In contrast to the conventional picture from phase-ordering kinetics, we find: (i) Domains at long time are highly interpenetrating and topologically complex, with average genus growing algebraically with system size. (ii) The long-time state is almost never static, but rather contains "blinker" spins that can flip ad infinitum with no energy cost. (iii) The energy relaxation has a complex time dependence with multiple characteristic time scales, the longest of whi…
Using Monte Carlo simulations based on the Metropolis algorithm, we investigate the dynamic phase transition properties of kinetic Ising model driven by a sinusoidally oscillating magnetic field in the presence of additive white noise. We calculate equilibrium and dynamic properties such as the temperature dependence of average magnetization and magnetic specific heat, as well as the period dependence of dynamic order parameter and scaled variance. After determining the critical period at which order-disorder transition takes place, we perform finite size scaling analysis to extract the expone…
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)…
We study the effects of next-nearest neighbor (NNN) interaction on the dynamic phase transition (DPT) and hysteresis loop area law in the two-dimensional ferromagnetic kinetic Ising model. We find that inclusion of the NNN interaction causes the DPT boundary of the NN kinetic Ising model to shift to larger values of magnetic field and temperature. The NNN kinetic Ising model can therefore exhibit an interaction induced DPT. Also in the low frequency limit (f$\to$0) the hysteresis loop area law, A(h$_{o}$,f), changes from h$^{0.70}_{o}$f$^{0.36}$ (NN) to h$^{0.14\pm 0.01}_{o}$f$^{0.13\pm0.01}$ …
We study the surface phase diagram of the three-dimensional kinetic Ising model below the equilibrium critical point subjected to a periodically oscillating magnetic field. Changing the surface interaction strength as well as the period of the external field, we obtain a non-equilibrium surface phase diagram that in parts strongly resembles the corresponding equilibrium phase diagram, with an ordinary transition, an extraordinary transition and a surface transition. These three lines meet at a special transition point. For weak surface couplings, however, the surface does not order. These resu…
We present evidence for a deep connection between the zero-temperature coarsening of the two-dimensional kinetic Ising model (KIM) and critical continuum percolation. In addition to reaching the ground state, the KIM can also fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly-solved critical percolation crossing probability P_{a,b} for a spanning path with winding numbers a and b.
The nonequilibrium dynamic phase transition, in the kinetic Ising model in presence of an oscillating magnetic field, has been studied both by Monte Carlo simulation (in two dimension) and by solving the meanfield dynamical equation of motion for the average magnetization. The temperature variations of hysteretic loss (loop area) and the dynamic correlation have been studied near the transition point. The transition point has been identified as the minimum-correlation point. The hysteretic loss becomes maximum above the transition point. An analytical formulation has been developed to analyse …
We introduce a driven version of the 1D kinetically constrained spin chain. In its original undriven version, this model shows anomalous coarsening following a quench to a low temperature, with an equilibration time that diverges as exp(1/T^2) for T\to 0. We show that driving of constant rate gamma_dot interrupts coarsening and stabilises the chain in a state analogous to that of a coarsening chain of age 1/gamma_dot. We present an analytical theory for this steady state, and demonstrate it to be in excellent agreement with our simulation results.
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 co…
The Glauber model on a one-dimensional lattice with boundaries (for the ferromagnetic- and anti-ferromagnetic case) is considered. The large-time behaviour of the one-point function is studied. It is shown that, for any positive temperature, the system shows a dynamical phase transition. The dynamical phase transition is controlled by the rate of spin flip at the boundaries, and is a discontiuous change of the derivative of the relaxation time towards the stationary configuration.
We perform extensive Monte Carlo simulations to investigate the dynamic phase transition properties of the two-dimensional kinetic Ising model on the kagome lattice in the presence of square-wave oscillating magnetic field. Through detailed finite-size scaling analysis, we study universality aspects of the non-equilibrium phase transition. Obtained critical exponents indicate that the two-dimensional kagome-lattice kinetic Ising model belongs to the same universality class with the corresponding Ising model in equilibrium. Moreover, dynamic critical exponent of the local moves used in simulati…
A spatially one dimensional coupled map lattice possessing the same symmetries as the Miller Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behaviour are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Is…
We study, using Monte Carlo dynamics, the time ($t$) dependent average magnetization per spin $m(t)$ behavior of 2-D kinetic Ising model under a binary ($\pm h_0$) stochastic field $h(t)$. The time dependence of the stochastic field is such that its average over each successive time interval $τ$ is assured to be zero (without any fluctuation). The average magnetization $Q=(1/τ)\int_{0}^τ m(t) dt$ is considered as order parameter of the system. The phase diagram in ($h_0,τ$) plane is obtained. Fluctuations in order parameter and their scaling properties are studied across the phase boundary. Th…
We study phase ordering dynamics in the three-dimensional nearest-neighbor Ising model, following rapid quenches from infinite to zero temperature. Results on various aspects, viz., domain growth, persistence, aging and pattern, have been obtained via the Glauber Monte Carlo simulations of the model on simple cubic lattice. These are analyzed via state-of-the-art methods, including the finite-size scaling, and compared with those for quenches to a temperature above the roughening transition. Each of these properties exhibit remarkably different behavior at the above mentioned final temperature…
We study low-temperature nucleation in kinetic Ising models by analytical and simulational methods, confirming the general result for the average metastable lifetime, <tau> = A*exp(beta*Gamma) (beta = 1/kT) [E. Jordao Neves and R.H. Schonmann, 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 doe…
Kinetic Ising models are powerful tools for studying the non-equilibrium dynamics of complex systems. As their behavior is not tractable for large networks, many mean-field methods have been proposed for their analysis, each based on unique assumptions about the system's temporal evolution. This disparity of approaches makes it challenging to systematically advance mean-field methods beyond previous contributions. Here, we propose a unifying framework for mean-field theories of asymmetric kinetic Ising systems from an information geometry perspective. The framework is built on Plefka expansion…
When periodically driven by an external magnetic field, a spin system can enter a phase of steady entrained oscillations with nonequilibrium probability distribution function. We consider an arbitrary magnetic field switching its direction with frequency comparable with the spin-flip rate and show that the resulting nonequilibrium probability distribution can be related to the system equilibrium distribution in the presence of a constant magnetic field of the same magnitude. We derive convenient approximate expressions for this exact relation and discuss their implications.
We analyze a kinetic Ising model with suppressed bulk noise which is a prominent representative of the generalized voter model phase transition. On the one hand we discuss the model in the context of social systems, and opinion formation in the presence of a tunable social temperature. On the other hand we characterize the abrupt phase transition. The system shows non-equilibrium dynamics in the presence of absorbing states. We slightly change the system to get a stationary state model variant exhibiting the same kind of phase transition. Using a Fokker-Planck description and comparing to mean…
We have examined the stationary state solutions of a bond diluted kinetic Ising model under a time dependent oscillating magnetic field within the effective-field theory (EFT) for a honeycomb lattice $(q=3)$. Time evolution of the system has been modeled with a formalism of master equation. The effects of the bond dilution, as well as the frequency $(ω)$ and amplitude $(h/J)$ of the external field on the dynamic phase diagrams have been discussed in detail. We have found that the system exhibits the first order phase transition with a dynamic tricritical point (DTCP) at low temperature and hig…
The one-dimensional Ising model is easily generalized to a \textit{genuinely nonequilibrium} system by coupling alternating spins to two thermal baths at different temperatures. Here, we investigate the full time dependence of this system. In particular, we obtain the evolution of the magnetisation, starting with arbitrary initial conditions. For slightly less general initial conditions, we compute the time dependence of all correlation functions, and so, the probability distribution. Novel properties, such as oscillatory decays into the steady state, are presented. Finally, we comment on the …
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 or…
An extension of the Kinetic Ising model with nonuniform coupling constants on a one-dimensional lattice with boundaries is investigated, and the relaxation of such a system towards its equilibrium is studied. Using a transfer matrix method, it is shown that there are cases where the system exhibits a dynamical phase transition. There may be two phases, the fast phase and the slow phase. For some region of the parameter space, the relaxation time is independent of the reaction rates at the boundaries. Changing continuously the reaction rates at the boundaries, however, there is a point where th…
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 agree…
