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arXiv:cond-mat/9702203 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Non-Markovian persistence and nonequilibrium critical dynamics

Klaus Oerding, Stephen J. Cornell, Alan J. Bray

Published Jul 1, 1997• 62 citations

Abstract

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)].

DOI: 10.1103/physreve.56.r25

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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)].

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 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.

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.