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.
Finite-size scaling (FSS) for a critical phase transition (
t=0) states that within a window of size
∣t∣∼L−1/ν, the scaling behavior of any observable
Q in a system of linear size
L asymptotically follows a scaling form as
Q(t,L)=LYQQ~(tL1/ν), where
ν is the correlation-length exponent,
YQ is an FSS exponent and
Q~(x) is a function of the scaled distance-to-criticality
x≡tL1/ν. We systematically study the asymptotic scaling behavior of
Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with
t→0 and
∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as
∣t∣∼L−λ with
λ<1/ν, the FSS becomes
λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality
t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
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.
A ``persistence exponent'' $\ensuremath{\theta}$ is defined for nonequilibrium critical phenomena. It describes the probability, $p(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}$, that the global order parameter has not changed sign in the time interval
t following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ limit of the
O(n) model, to first order in $\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d$, and for the 1D Ising model. Numerical results are obtained for the 2D Ising model. We argue that $\ensuremath{\theta}$ is a new independent exponent.
The persistence exponent $\ensuremath{\theta}$ for the global order parameter
M(t) of a system quenched from the disordered phase to its critical point describes the probability, $p(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}$, that
M(t) does not change sign in the time interval
t following the quench. We calculate $\ensuremath{\theta}$ to $O({\ensuremath{\epsilon}}^{2})$ for model
A of Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)] (and to order $\ensuremath{\epsilon}$ for model
C) and show that at this order
M(t) is a non-Markov process. Consequently, to our knowledge, $\ensuremath{\theta}$ is a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Majumdar and Sire [Phys. Rev. Lett. 77, 1420 (1996)].
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.