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arXiv:cond-mat/0108438 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Pair contact process with a particle source

Ronald Dickman, WILSON Ricardo Matos Rabelo, Géza Ódor

Published Dec 18, 2001• 18 citations
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Abstract

We study the phase diagram and critical behavior of the one-dimensional pair contact process (PCP) with a particle source using cluster approximations and extensive simulations. The source creates isolated particles only, not pairs, and so couples not to the order parameter (the pair density) but to a nonordering field, whose state influences the evolution of the order parameter. While the critical point p(c) shows a singular dependence on the source intensity, the critical exponents appear to be unaffected by the presence of the source, except possibly for a small change in beta. In the course of our paper, we obtain high-precision values for the critical exponents of the standard PCP, confirming directed-percolationlike scaling.

DOI
10.1103/physreve.65.016118

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We study the phase diagram and critical behavior of the one-dimensional pair contact process (PCP) with a particle source using cluster approximations and extensive simulations. The source creates isolated particles only, not pairs, and so couples not to the order parameter (the pair density) but to a nonordering field, whose state influences the evolution of the order parameter. While the critical point p(c) shows a singular dependence on the source intensity, the critical exponents appear to be unaffected by the presence of the source, except possibly for a small change in beta. In the course of our paper, we obtain high-precision values for the critical exponents of the standard PCP, confirming directed-percolationlike scaling.
The one-dimensional pair contact process with a particle source is studied by using dynamical cluster mean-field approximations with sites up to n=12. The results obtained for different levels of approximation become convergent especially for n>or=6 and allow us to derive reliable extrapolations to the limit n--> infinity. At the zero source limit, the critical point exhibits a discontinuity whose magnitude vanishes with 1/n. Coherent anomaly analysis of the data supports the conclusion that the vanishing of the order parameter and the density of isolated particles have the same critical behavior. In contrast to an earlier prediction, the present approximation does not support the existence of critical behavior in the inactive phase where the frozen density of isolated particles depends on the initial state.
The phase transition of the one-dimensional, diffusive pair contact process (PCPD) is investigated by N cluster mean-field approximations and high precision simulations. The N=3,4 cluster approximations exhibit smooth transition line to absorbing state by varying the diffusion rate D with beta_2=2 mean-field order parameter exponent of the pair density. This contradicts with former N=2 results, where two different mean-field behavior was found along the transition line. Extensive dynamical simulations on L=10^5 lattices give estimates for the order parameter exponents of the particles for 0.05 <= D <= 0.7. These data can support former two distinct class findings. However the gap between low and high D exponents is narrower than estimated previously and the possibility for interpreting numerical data as a single class behavior with exponents alpha=0.21(1), beta=0.41(1) assuming logarithmic corrections is shown. Finite size scaling and cluster simulation results are also presented.
The pair contact process with diffusion (PCPD) is studied with a standard Monte Carlo approach and with simulations at fixed densities. A standard analysis of the simulation results, based on the particle densities or on the pair densities, yields inconsistent estimates for the critical exponents. However, if a well-chosen linear combination of the particle and pair densities is used, leading corrections can be suppressed, and consistent estimates for the independent critical exponents delta=0.16(2) , beta=0.28(2) , and z=1.58 are obtained. Since these estimates are also consistent with their values in directed percolation (DP), we conclude that the PCPD falls in the same universality class as DP.
The pair contact process (PCP) is a nonequilibrium stochastic model which, like the basic contact process (CP), exhibits a phase transition to an absorbing state. The two models belong to the directed percolation (DP) universality class, despite the fact that the PCP possesses infinitely many absorbing configurations whereas the CP has but one. The critical behavior of the PCP with hopping by particles (PCPD) is as yet unclear. Here we study a version of the PCP in which nearest-neighbor particle pairs can hop but individual particles cannot. Using quasistationary simulations for three values of the diffusion probability ( D = 0.1, 0.5 and 0.9), we find convincing evidence of DP-like critical behavior.
The cluster mean-field approximations are performed, up to 13 cluster sizes, to study the critical behavior of the driven pair contact process with diffusion (DPCPD) and its precedent, the PCPD in one dimension. Critical points are estimated by extrapolating our data to the infinite cluster size limit, which are in good accordance with recent simulation results. Within the cluster mean-field approximation scheme, the PCPD and the DPCPD share the same mean-field critical behavior. The application of the coherent anomaly method, however, shows that the two models develop different coherent anomalies, which lead to different true critical scaling. The values of the critical exponents for the particle density, the pair density, the correlation length, and the relaxation time are fairly well estimated for the DPCPD. These results support and complement our recent simulation results for the DPCPD.
We study the pair contact process with diffusion (PCPD) using Monte Carlo simulations, and concentrate on the decay of the particle density ρ with time, near its critical point, which is assumed to follow . This model is known for its slow convergence to the asymptotic critical behavior; we therefore pay particular attention to finite-time corrections. We find that at the critical point, the ratio of ρ and the pair density ρ p converges to a constant, indicating that both densities decay with the same power law. We show that under the assumption δ 2 ≈2δ, two of the critical exponents of the PCPD model are δ = 0.165(10) and β = 0.31(4), consistent with those of the directed percolation (DP) model.