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arXiv:0907.1642 · Journal of Statistical Mechanics Theory and Experiment

Non-Markovian global persistence in phase ordering kinetics

Malte Henkel, Michel Pleimling

Published Dec 21, 2009• 6 citations
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Abstract

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.

DOI
10.1088/1742-5468/2009/12/p12012

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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.
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.
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 ttt 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)O(n)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)M(t)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)M(t)M(t) does not change sign in the time interval ttt following the quench. We calculate $\ensuremath{\theta}$ to $O({\ensuremath{\epsilon}}^{2})$ for model AAA of Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)] (and to order $\ensuremath{\epsilon}$ for model CCC) and show that at this order M(t)M(t)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)].
One-dimensional nonequilibrium 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 and the exponent characterizing the two-time autocorrelation function of the total magnetization under nonequilibrium conditions are reported. It is found that the critical fluctuations at the PC transition have a strong effect on the spins: the behaviour becomes non-Markovian and the above exponents exhibit drastic changes as compared with the Markovian Glauber - Ising case. In this context the crucial importance of considering the global order parameter (instead of the local one) is emphasized.
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.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.