We investigate the persistence properties of critical d -dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d '-dimensional manifold. The persistence probability P c ( t ) shows three distinct long-time decays depending on the value of the parameter ζ = ( D − 2 + η)/ z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d − d ' and of the critical exponents η and z . We find that the asymptotic behavior of P c ( t ) is exponential for ζ > 1, stretched exponential for 0 ≤ ζ ≤ 1, and algebraic for ζ < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d ' = d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising models, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O( n ) model in the limit and its subtle connection with the spherical model are also discussed in detail. In particular, we show that the correlation functions of the components of the order parameter which are respectively parallel and transverse to its average value within the model correspond to the correlation functions of the local and global order parameters of the spherical model.
We investigate the persistence properties of critical d -dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d '-dimensional manifold. The persistence probability P c ( t ) shows three distinct long-time decays depending on the value of the parameter ζ = ( D − 2 + η)/ z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d − d ' and of the critical exponents η and z . We find that the asymptotic behavior of P c ( t ) is exponential for ζ > 1, stretched exponential for 0 ≤ ζ ≤ 1, and algebraic for ζ < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d ' = d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising models, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O( n ) model in the limit and its subtle connection with the spherical model are also discussed in detail. In particular, we show that the correlation functions of the components of the order parameter which are respectively parallel and transverse to its average value within the model correspond to the correlation functions of the local and global order parameters of the spherical model.
We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter ( e.g. , the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m 0 of the order parameter and the typical time at which it occurs diverges as m 0 vanishes. Monte Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.
We study the persistence probability P(t) that, starting from a random initial condition, the magnetization of a ${d}^{\ensuremath{'}}$-dimensional manifold of a d-dimensional spin system at its critical point does not change sign up to time t. For ${d}^{\ensuremath{'}}>0$ we find three distinct late-time decay forms for P(t): exponential, stretched exponential, and power law, depending on a single parameter $\ensuremath{\zeta}=(D\ensuremath{-}2+\ensuremath{\eta})/z$, where $D=d\ensuremath{-}{d}^{\ensuremath{'}}$ and $\ensuremath{\eta},z$ are standard critical exponents. In particular, we predict that for a line magnetization in the critical d=2 Ising model, P(t) decays as a power law while, for d=3, P(t) decays as a power of t for a plane magnetization but as a stretched exponential for a line magnetization. Numerical results are consistent with these predictions.
We investigate global persistence properties for the non-equilibrium critical dynamics of the randomly diluted Ising model. The disorder-averaged persistence probability c ( t ) of the global magnetization is found to decay algebraically with an exponent θ c that we compute analytically in a dimensional expansion in d = 4 − . Corrections to Markov process are found to occur already at one loop order and θ c is thus a novel exponent characterizing this disordered critical point. Our result is thoroughly compared with Monte Carlo simulations in d = 3, which also include a measurement of the initial slip exponent. Taking carefully into account corrections to scaling, θ c is found to be universal, independent of the dilution factor p along the critical line at T c ( p ), and in good agreement with our one-loop calculation.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions d=2 and 3. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (Tf) below the critical one (Tc), from different initial temperatures Ti≥Tc. In long time limit, for Ti>Tc, the persistence probability exhibits power-law decay with exponents θ≃0.22 and ≃0.18 in d=2 and 3, respectively. For finite Ti, the early time behavior is a different power-law whose life-time diverges and exponent decreases as Ti→Tc. The crossover length between the two steps diverges as the equilibrium correlation length. Ti=Tc is expected to provide a {\it{new universality class}} for which we obtain θ≃0.035 in d=2 and ≃0.10 in d=3. The time dependence of the average domain size ℓ, however, is observed to be rather insensitive to the choice of Ti.
The persistence probability P g ( t ) of the global order parameter of a simple ferromagnet undergoing phase ordering kinetics after a quench from a fully disordered state to below the critical temperature, T < T c , is analysed. It is argued that the persistence probability decays algebraically with time in the entire low-temperature phase. For Markov processes, the associated global persistence exponent θ g = (2λ C − d )/(2 z ) is related to the autocorrelation exponent λ C . This relationship is confirmed for phase ordering in the exactly solved 1D Ising model and the d -dimensional spherical model. For the 2D Glauber–Ising model, the temperature-independent estimate θ g = 0.063(2) indicates that the dynamics of the global order parameter is described by a non-Markovian process.