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arXiv:cond-mat/0010021 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Numerical study of persistence in models with absorbing states

Ezequiel Vicente Albano, Miguel A. Muñoz

Published Feb 21, 2001• 34 citations

Abstract

Extensive Monte Carlo simulations are performed in order to evaluate both the local (straight theta(l)) and global (straight theta(g)) persistence exponents in the Ziff-Gulari-Barshad (ZGB) [Phys. Rev. Lett. 56, 2553 (1986)] irreversible reaction model. At the second-order irreversible phase transition (IPT) we find that both the local and the global persistence exhibit power-law behavior with a crossover between two different time regimes. On the other hand, at the ZGB first-order IPT, active sites are short lived and the persistence decays more abruptly; it is not clear whether it shows power-law behavior or not. In order to analyze universality issues, we have also studied another model with absorbing states, the contact process, and evaluated the local persistence exponent in dimensions from 1 to 4. A striking apparent superuniversality is reported: the local persistence exponent seems to coincide in both one- and two-dimensional systems. Some other aspects of persistence in systems with absorbing states are also analyzed.

DOI: 10.1103/physreve.63.031104

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Extensive Monte Carlo simulations are performed in order to evaluate both the local (straight theta(l)) and global (straight theta(g)) persistence exponents in the Ziff-Gulari-Barshad (ZGB) [Phys. Rev. Lett. 56, 2553 (1986)] irreversible reaction model. At the second-order irreversible phase transition (IPT) we find that both the local and the global persistence exhibit power-law behavior with a crossover between two different time regimes. On the other hand, at the ZGB first-order IPT, active sites are short lived and the persistence decays more abruptly; it is not clear whether it shows power-law behavior or not. In order to analyze universality issues, we have also studied another model with absorbing states, the contact process, and evaluated the local persistence exponent in dimensions from 1 to 4. A striking apparent superuniversality is reported: the local persistence exponent seems to coincide in both one- and two-dimensional systems. Some other aspects of persistence in systems with absorbing states are also analyzed.

We studied the continuous phase transition between the active and the absorbing state of the Ziff-Gulari-Barshad (ZGB) model. Through Monte Carlo simulations we determined all the moments of the order parameter up to fourth order and their ratios at the critical point. We show that the ratios we found are in agreement with those of the contact and pair contact processes in two dimensions, which give support to the idea that the ZGB model is in the directed percolation universality class in (2+1) dimensions.

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 time dependence of the local persistence probability during a nonstationary time evolution in the disordered contact process in d=1, 2, and 3 dimensions. We present a method for calculating the persistence with the strong-disorder renormalization group (SDRG) technique, which we then apply at the critical point analytically for d=1 and numerically for d=2,3. According to the results, the average persistence decays at late times as an inverse power of the logarithm of time, with a universal dimension-dependent generalized exponent. For d=1, the distribution of sample-dependent local persistence is shown to be characterized by a universal limit distribution of effective persistence exponents. Using a phenomenological approach of rare-region effects in the active phase, we obtain a nonuniversal algebraic decay of the average persistence for d=1 and enhanced power laws for d>1. As an exception, for randomly diluted lattices, the algebraic decay remains valid for d>1, which is explained by the contribution of dangling ends. Results on the time dependence of average persistence are confirmed by Monte Carlo simulations. We also prove the equivalence of the persistence with a return probability, a valuable tool for the argumentations.

The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).

I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. Time-dependent Monte Carlo simulations in the vicinity of the critical point yield very accurate estimates for the location of the critical point and several critical exponents. Time-dependent simulations in the supercritical and subcritical regimes yield estimates for further critical exponents from the asymptotic behavior of the survival probability and the average number of particles. The results in the super- and subcritical regimes are also analyzed by utilizing the scaling laws describing the behavior of the model. The results for the critical exponents are the best obtained so far for a three-dimensional model with directed-percolation critical behavior.