The two dimensional ferromagnetic Ising model in the presence of a propagating magnetic field wave (with well defined frequency and wavelength) is studied by Mone Carlo simulation. This study differs from all of the earlier studies done so far, where the oscillating magnetic field was considered to be uniform in space. The time average magnetisation over a full cycle (the time period) of the propagating magnetic field acts as the dynamic order parameter. The dynamical phase transition is observed. The temperature variation of the dynamic order parameter, the mean square deviation of the dynamic order parameter, the dynamic specific heat and the derivative of the dynamic order parameter are studied. The mean square deviation of the dynamic order parameter, dynamic specific heat show sharp maxima near the transition point. The derivative of dynamic order parameter shows sharp minimum near the transition point. The transition temperature is found to depend also on the speed of propagation of the magnetic field wave.
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.
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.
We investigate nonequilibrium relaxations of Ising models at the critical point by using a cluster update. While preceding studies imply that nonequilibrium cluster-flip dynamics at the critical point are universally described by the stretched-exponential function, we find that the dynamics changes from the stretched-exponential to the power function as the dimensionality is increased: The two-, three-, four-, and infinite-dimensional Ising models are numerically studied, and the four- and infinite-dimensional Ising models exhibit the power-law relaxation. We also show that the finite-size sca…
In this paper, we review the physical concepts of the nonequilibrium techniques for the calculation of free energies applied to magnetic systems using Monte Carlo simulations of different nonequilibrium processes. The methodology allows the calculation of the free energy difference between two different system Hamiltonians, as well as the free energy dependence on temperature and magnetic field for a given Hamiltonian. As an illustration of the effectiveness of this approach, we apply the methodologies to determine the phase diagram of a simple microscopic model, the degenerate Ising model. Ou…
Nonequilibrium wetting transitions are observed in Monte Carlo simulations of a kinetic spin system in the absence of a detailed balance condition with respect to an energy functional. A nonthermal model is proposed starting from a two-dimensional Ising spin lattice at zero temperature with two boundaries subject to opposing surface fields. Local spin excitations are only allowed by absorbing an energy quantum (photon) below a cutoff energy E_c. Local spin relaxation takes place by emitting a photon which leaves the lattice. Using Monte Carlo simulation nonequilibrium critical wetting transiti…
We studied the nonequilibrium magnetisation reversal in kinetic Ising ferromagnets driven by periodic impulsive magnetic field in meanfield approximation. The meanfield differential equation was solved by sixth order Runge-Kutta-Felberg method. The periodicity and strength of the applied impulsive magnetic field play the key role in magnetisation reversal. We studied the minimum strength of impulsive field required for magnetisation reversal at any temperature as a function of the periodicity of the impulsive field. In the high temperature and small period this is observed to be linear. The re…
Recently Mayer et al. [Phys. Rev. E {\bf 68}, 016116 (2003)] proposed a new way to compute numerically the fluctuation-dissipation ratios in nonequilibrium critical systems. Using well-known facts of nonequilibrium critical dynamics I show that the leading contributions of the quantities they consider are in fact one-time quantities which are independent of the waiting time. The ratio of these one-time quantities determines the slope of the straight lines observed in the fluctuation-dissipation plots of Mayer et al.
The dynamical responses of a rodlike anisotropic Heisenberg ferromagnet, irradiated by the propagating magnetic field wave, have been studied by Monte Carlo simulation using Metropolis single spin-flip algorithm. A nonequilibrium dynamical phase transition has been observed. The transition temperature has been obtained from the peak position of the variance of the dynamic order parameter plotted as a function of the temperature. The transition has been found to occur at a lower temperature for the higher value of the amplitude of the propagating magnetic field wave. A comprehensive phase bound…
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 study domain distributions in the one-dimensional Ising model subject to zero-temperature Glauber and Kawasaki dynamics. The survival probability of a domain, $S(t)\sim t^{-ψ}$, and an unreacted domain, $Q_1(t)\sim t^{-δ}$, are characterized by two independent nontrivial exponents. We develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents.
While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations $p$ and $1-p$, respectively. Although in our model for $p<1$ a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any $p>0$ is shown to satisfy the same equation as for the pure Ising model ($p=1$). Numerical simulations confirm such a behavior, but the equivalence with the pure Ising model apparently h…
The nonequilibrium responses of Ising metamagnet (layered antiferromagnet) to the propagating magnetic wave are studied by Monte Carlo simulation. Here, the spatio-temporal variations of magnetic field keeps the system away from equilibrium. The sublattice magnetisations show dynamical symmetry breaking in the low temperature ordered phase. The nonequilibrium phase transitions are studied from the temperature dependences of dynamic staggered order parameter, its derivative and the dynamic specific heat. The transitions are marked by the peak position of dynamic specific heat and the position o…
The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on …
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 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 study the nonequilibrium steady states of an Ising chain in a transverse field, h, by investigating the effect of a field, lambda, which drives the current of energy. The zero-temperature, h-lambda phase diagram is determined exactly and it is found that the energy current appears continuously above a threshold, lambda > lambda_c(h). The long-range magnetic order (present for h<h_c, lambda < lambda_c) is destroyed in the current-carrying state where the correlations are characterized by power-law, oscillatory decay. The mechanism which generates the power-law correlations in the current-car…
The dynamical steady state behaviour of the random field Ising ferromagnet swept by a propagating magnetic field wave is studied at zero temperature by Monte Carlo simulation in two dimensions. The distribution of the random field is bimodal type. For a fixed set of values of the frequency and wavelength of propagating magnetic field wave and the strength of the random field, four distinct dynamical steady states or nonequilibrium phases were identified. These four nonequilibrium phases are characterised by different values of structure factors. State of first kind, where all spins are paralle…
We investigate a nonequilibrium coarsening dynamics of a one-dimensional Ising spin system with chirality. Only spins at domain boundaries are updated so that the model undergoes a coarsening to either of equivalent absorbing states with all spins + or -. Chirality is imposed by assigning different transition rates to events at down (+-) kinks and up (-+) kinks. The coarsening is characterized by power-law scalings of the kink density $ρ\sim t^{-δ}$ and the characteristic length scale $ξ\sim t^{1/z}$ with time $t$. Surprisingly the scaling exponents vary continuously with model parameters, whi…
The nonequilibrium behaviour of a core-shell nanoparticle has been studied by Monte- Carlo simulation. The core consists of Ising spins of $σ=1/2$ and the shell contains Ising spins of $S=1$. The interactions within the core and in the shell are considered ferromagnetic but the interfacial interaction between core and shell is antiferromagnetic. The nanoparticle system is kept in open boundary conditions and is driven by randomly varying (in time but uniform over the space) magnetic field. Depending on the width of the randomly varying field and the temperature of the system, the core, shell a…
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time t and by Monte Carlo simulation. Both the magnetization (m) and energy series are obtained up to 12-th order. An accurate estimate from series analysis for the dynamical critical exponent z is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, z_eff(t) = - {1/8} d ln t /d ln m, directly from a ratio of three-spin correlation to m. Extrapolation to t = infinity leads to an estimate z = 2.169 +/- 0.003.
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)…
The finite size analysis of the nonequilibrium phase transition, in two dimensional Ising ferromagnet driven by plane propagating magneticwave, is studied by Monte Carlo simulation. It is observed that the system undergoes a nonequilibrium dynamicphase transition from a high temperature dynamically symmetric ({\it propagating}) phase to a low temperature dynamically symmetry-broken ({\it pinned}) phase as the system is cooled below the transition temperature. This transition temperature is determined precisely by studying the fourth order Binder Cumulant of the dynamic order parameter as a fun…
The dynamical responses of XY ferromagnet driven by linearly polarised propagating and standing magnetic field wave have been studied by Monte Carlo simulation in three dimensions. In the case of propagating magnetic field wave (with specified amplitude, frequency and the wavelength), the low temperature dynamical mode is a propagating spin wave and the system becomes structureless (or random) in the high temperature. A dynamical symmetry breaking phase transition is observed at a finite (nonzero) temperature. This symmetry breaking is confirmed by studying the statistical distribution of the …
The dynamical responses of Blume-Capel (S=1) ferromagnet to the plane propagating (with fixed frequency and wavelength) and standing magnetic field waves are studied sepa- rately in two dimensions by extensive Monte Carlo simulation. Depending on the values of temperature, amplitude of the propagating magnetic field and the strength of anisotropy, two different dynamical phases are observed. For a fixed value of anisotropy and the amplitude of the propagating magnetic field, the system undergoes a dynamical phase transition from a driven spin wave propagating phase to a pinned or spin frozen s…
The Binder ratios exhibit discrepancy from the Gaussian behavior of the magnetic cumulants, and their size independence at the critical point has been widely utilized in numerical studies of critical phenomena. In the present article we reformulate the nonequilibrium relaxation (NER) analysis in cluster algorithms using the $(2,1)$-Binder ratio, and apply this scheme to the two- and three-dimensional Ising models. Although the stretched-exponential relaxation behavior at the critical point is not explicitly observed in this quantity, we find that there exists a logarithmic finite-size scaling …
We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field of strength, $h=H T k_B$, as well as the Villain model on the square lattice. After a quench from a disordered initial state to T=0 we study the nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linear size, $L$, we define and measure sample dependent "first passage time", $t_r$, which is the number of Monte Carlo steps until the energy is relaxed to the ground-state value. The distribution of $t_r$, in particular its mean value, $< t_r(L) >$, is shown to obey…
Nonequilibrium behavior and dynamic phase transition properties of a kinetic Ising model under the influence of periodically oscillating random-fields have been analyzed within the framework of effective field theory (EFT) based on a decoupling approximation (DA). Dynamic equation of motion has been solved for a simple cubic lattice ($q=6$) by utilizing a Glauber type stochastic process. Amplitude of the sinusoidally oscillating magnetic field is randomly distributed on the lattice sites according to bimodal and trimodal distribution functions. For a bimodal type of amplitude distribution, it …
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 determi…
We investigate the phase transitions in a coupled system of Ising spins and a fluctuating network. Each spin interacts with $q$ neighbors through links of the rewiring network. The Ising spins and the network are in thermal contact with the heat baths at temperatures $T_S$ and $T_L$, respectively, so that the whole system is driven out of equilibrium for $T_S \neq T_L$. The model is a generalization of the $q$-neighbor Ising model, which corresponds to the limiting case of $T_L=\infty$. Despite the mean field nature of the interaction, the $q$-neighbor Ising model was shown to display a discon…
We investigate the temporal evolution of a ferromagnetic system of Ising spins evolving under Kawasaki dynamics from a random initial condition, in spatial dimensions one and two. We examine in detail the asymptotic behaviour of the two-time correlation and response functions. The linear response is measured without applying a field, using a recently proposed algorithm. For the chain at vanishingly small temperature, we introduce an accelerated dynamics which has the virtue of projecting the system into the asymptotic scaling regime. This allows us to revisit critically previous works on the b…
The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical properties of the quantum chain. The remaining part is devoted to the nonequilibrium dynamical behaviour of such quantum c…
We present a numerical study on the ordering dynamics of a one-dimensional nonequilibrium Ising spin system with chirality. This system is characterized by a direction-dependent spin update rule. Pairs of $+-$ spins can flip to $++$ or $--$ with probability $(1-u)$ or to $-+$ with probability $u$ while $-+$ pairs are frozen. The system was found to evolve into the ferromagnetic ordered state at any $u<1$ exhibiting the power-law scaling of the characteristic length scale $ξ\sim t^{1/z}$ and the domain wall density $ρ\sim t^{-δ}$. The scaling exponents $z$ and $δ$ were found to vary continuousl…
We study the nonequilibrium critical point of the zero temperature random field Ising model on a triangular lattice and compare it with known results on honeycomb, square, and simple cubic lattices. We suggest that the coordination number of the lattice rather than its dimension plays the key role in determining the universality class of the nonequilibrium critical behavior. This is discussed in the context of numerical evidence that equilibrium and nonequilibrium critical points of the zero-temperature random field Ising model belong to the same universality class. The physics of this curious…
We study critical hysteresis in the random-field Ising model (RFIM) on a two-dimensional periodic lattice with a variable coordination number $z_{eff}$ in the range $3 \le z_{eff} \le 6$. We find that the model supports critical behavior in the range $4 < z_{eff} \le 6$, but the critical exponents are independent of $z_{eff}$. The result is discussed in the context of the universality of nonequilibrium critical phenomena and extant results in the field.
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.
The nonequilibrium dynamic phase transitions in ferromagnetic models (Ising, XY and Heisenberg) are reviewed on the basis of very recent work in this field.
We studied the critical behavior of the $J_{1}-J_{2}$ spin-{1/2} Ising model in the square lattice by considering $J_{1}$ fixed and $J_{2}$ as random interactions following discrete and continuous probability distribution functions. The configuration of $J_{2}$ in the lattice evolves in time through a competing kinetics using Monte Carlo simulations leading to a steady state without reaching the free-energy minimization. However, the resulting non-equilibrium phase diagrams are, in general, qualitatively similar to those obtained with quenched randomness at equilibrium in past works. According…
Nonequilibrium phase transition properties of a mixed Ising ferrimagnetic model consisting of spin-1/2 and spin-3/2 on a square lattice under the existence of a time dependent oscillating magnetic field have been investigated by making use of Monte Carlo simulations with single-spin flip Metropolis algorithm. A complete picture of dynamic phase boundary and magnetization profiles have been illustrated and the conditions of a dynamic compensation behavior have been discussed in detail. According to our simulation results, the considered system does not point out a dynamic compensation behavior,…
The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension $d_H$ =1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. The topological effects become evident from the emergence of a logarithmic periodic oscillation superimposed to a…
The nonequilibrium dynamic phase transition in ferromagnetic systems is reviewed. Very recent results of dynamic transition in kinetic Ising model and that in Heisenberg ferromagnet is discussed.
We studied the dynamic response and stochastic resonance of kinetic Ising spin system (ISS), subject to the joint external field of weak sinusoidal modulation and stochastic white-noise, through solving the mean-field equation of motion based on Glauber dynamics. The periodically driven stochastic ISS shows the occurrence of characteristic stochastic resonance as well as nonequilibrium dynamic phase transition (NDPT) when the frequency and amplitude h0 of driving field, the temperature t of the system and noise intensity D attain a specific accordance in quantity. There exist in the system two…
The real part of the time-dependent ac susceptibility of the short-range Ising spin glass in a transverse field has been investigated at very low temperatures. We have used the quantum linear response theory and domain coarsening ideas of quantum droplet scaling theory. It is found that after a temperature quench to a temperature which is less than the spin glass transition temperature the ac susceptibility decreases with time elapsed after the initial quench towards equilibrium approximately logarithmically. It is shown that the transverse field of "tunneling" has nonessential effect to a non…
The non-equilibrium dynamics of the three-dimensional Edwards-Anderson spin-glass model with different bond distributions is investigated by means of Monte Carlo simulation. A numerical method is used to determine the critical temperature and the scaling exponents of the correlation and the integrated response functions. The results obtained agree with those calculated in equilibrium simulations and suggest that the universality class does not depend on the exact form of the bond distribution.
The finite-size scaling method in the equilibrium Monte Carlo(MC) simulations and the finite-time scaling method in the nonequilibrium-relaxation simulations are compromised. MC time data of various physical quantities are scaled by the MC time data of the dynamic correlation length, which corresponds to changing the system size in the finite-size scaling method. This scaling method is tested in the three-dimensional ferromagnetic Ising spin model and in the three dimensional $\pm J$ Ising spin-glass model. The transition temperature and the critical exponents, $η$ and $ν$, are obtained by the…
Using an Ising-like model of protein mechanical unfolding, we introduce a diffusive dynamics on its exactly known free energy profile, reducing the nonequilibrium dynamics of the model to a biased random walk. As an illustration, the model is then applied to the protein translocation phenomenon, taking inspiration from a recent experiment on the green fluorescent protein pulled by a molecular motor. The average translocation time is evaluated exactly, and the analysis of single trajectories shows that translocation proceeds through an intermediate state, similar to that observed in the experim…
In order to elucidate the role of surfaces at nonequilibrium phase transitions we consider kinetic Ising models with surfaces subjected to a periodic oscillating magnetic field. Whereas the corresponding bulk system undergoes a continuous nonequilibrium phase transition characterized by the exponents of the equilibrium Ising model, we find that the nonequilibrium surface exponents do not coincide with those of the equilibrium critical surface. In addition, in three space dimensions the surface phase diagram of the nonequilibrium system differs markedly from that of the equilibrium system.
The dynamical response of spin-S (S=1, 3/2, 2, 3) Ising ferromagnet to the plane propagating wave , standing magnetic field wave and uniformly oscillating field with constant frequency are studied separately in two dimensions by extensive Monte Carlo simulation. Depending upon the strength of the magnetic field and the value of the spin state of the Ising spin lattice two different dynamical phases are observed. For a fixed value of S and the amplitude of the propagating magnetic field wave the system undergoes a dynamical phase transition from propagating phase to pinned phase as the temperat…
We study the nonequilibrium properties of directed Ising models with non conserved dynamics, in which each spin is influenced by only a subset of its nearest neighbours. We treat the following models: (i) the one-dimensional chain; (ii) the two-dimensional square lattice; (iii) the two-dimensional triangular lattice; (iv) the three-dimensional cubic lattice. We raise and answer the question: (a) Under what conditions is the stationary state described by the equilibrium Boltzmann-Gibbs distribution? We show that for models (i), (ii), and (iii), in which each spin "sees" only half of its neighbo…
We investigate properties of the diffusive motion of an interface in the two-dimensional Ising model in equilibrium or nonequilibrium situations. We focused on the relation between the power spectrum of a time sequence of spins and diffusive motion of an interface which was already clarified in one- dimensional systems with a nonequilibrium phase transition like the asymmetric simple exclusion process. It is clarified that the interface motion is a diffusion process with a drift force toward the higher-temperature side when the system is in contact with heat reservoirs at different temperature…
We investigate the nonequilibrium dynamics following a quench to zero temperature of the non-conserved Ising model with power-law decaying long-range interactions $\propto 1/r^{d+σ}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $α$, the persistence exponent $θ$, and the fractal dimension $d_f$. It is found that the growth exponent $α\approx 3/4$ is independent of $σ$ and different from $α=1/2$ …
The dynamical response of an Ising ferromagnet to a plane polarised standing magnetic field wave is modelled and studied here by Monte Carlo simulation in two dimensions. The amplitude of standing magnetic wave is modulated along the direction x. We have detected two main dynamical phases namely, pinned and oscillating spin clusters. Depending on the value of field amplitude the system is found to undergo a phase transition from oscillating spin cluster to pinned as the system is cooled down. The time averaged magnetisation over a full cycle of magnetic field oscillations is defined as the dyn…
We study nonequilibrium dynamics of the quantum Ising chain at zero temperature when the transverse field is varied stochastically. In the equivalent fermion representation, the equation of motion of Majorana operators is derived in the form of a one-dimensional, continuous-time quantum random walk with stochastic, time-dependent transition amplitudes. This type of external noise gives rise to decoherence in the associated quantum walk and the semiclassical wave-packet generally has a diffusive behavior. As a consequence, in the quantum Ising chain, the average entanglement entropy grows in ti…
We consider the non-conserved dynamics of the Ising model on the two-dimensional square lattice, where each spin is influenced preferentially by its East and North neighbours. The single-spin flip rates are such that the stationary state is Gibbsian with respect to the usual ferromagnetic Ising Hamiltonian. We show the existence, in the paramagnetic phase, of a dynamical transition between two regimes of violation of the fluctuation-dissipation theorem in the nonequilibrium stationary state: a regime of weak violation where the stationary fluctuation-dissipation ratio is finite, when the asymm…
With a nonequilibrium relaxation method, we calculate the dynamic critical exponent z of the two-dimensional Ising model for the Swendsen-Wang and Wolff algorithms. We examine dynamic relaxation processes following a quench from a disordered or an ordered initial state to the critical temperature T_c, and measure the exponential relaxation time of the system energy. For the Swendsen-Wang algorithm with an ordered or a disordered initial state, and for the Wolff algorithm with an ordered initial state, the exponential relaxation time fits well to a logarithmic size dependence up to a lattice si…
