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arXiv:cond-mat/9606123 · Physical Review Letters

Global Persistence Exponent for Nonequilibrium Critical Dynamics

Satya N. Majumdar, Alan J. Bray, Stephen J. Cornell, Clément Sire

Published Oct 28, 1996• 168 citations
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Abstract

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.

DOI
10.1103/physrevlett.77.3704

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A ``persistence exponent'' \ensuremathθ\ensuremath{\theta}\ensuremathθ is defined for nonequilibrium critical phenomena. It describes the probability, p(t)\ensuremath∼t\ensuremath−\ensuremathθp(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}p(t)\ensuremath∼t\ensuremath−\ensuremathθ, 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=\ensuremath∞n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}n=\ensuremath∞ limit of the O(n)O(n)O(n) model, to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d, and for the 1D Ising model. Numerical results are obtained for the 2D Ising model. We argue that \ensuremathθ\ensuremath{\theta}\ensuremathθ 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)].
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.
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 consider a directed percolation process at its critical point. The probability that the deviation of the global order parameter with respect to its average has not changed its sign between 0 and t decays with t as a power law. In space dimensions the global persistence exponent that characterizes this decay is while for d<4 its value is increased to first order in . Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to to first order in . The global persistence exponent is found to be a new and independent exponent. Finally we compare our results with existing simulations.
We study the persistence probability P(t)P(t)P(t) that, starting from a random initial condition, the magnetization of a ${d}^{\ensuremath{'}}$-dimensional manifold of a ddd-dimensional spin system at its critical point does not change sign up to time ttt. For ${d}^{\ensuremath{'}}>0$ we find three distinct late-time decay forms for P(t)P(t)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=2d=2d=2 Ising model, P(t)P(t)P(t) decays as a power law while, for d=3d=3d=3, P(t)P(t)P(t) decays as a power of ttt for a plane magnetization but as a stretched exponential for a line magnetization. Numerical results are consistent with these predictions.