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arXiv:cond-mat/0109024 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Static critical behavior in the inactive phase of the pair contact process

M C Marques, Mariana Aparecida de Jesus Castro Santos, José F. F. Mendes

Published Dec 14, 2001• 8 citations
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Abstract

Steady-state properties in the absorbing phase of the $1d$ pair contact process model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles ${\ensuremath{\rho}}_{1},$ approaches a stationary value that depends on the annihilation probability $(p);$ the deviation from its ``natural'' value at criticality ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}$ follows a power law: ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}\ensuremath{-}{\ensuremath{\rho}}_{1}\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}_{1}}$ for $p>{p}_{c}.$ Monte Carlo simulations yield ${\ensuremath{\beta}}_{1}=0.81.$ A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting ${\ensuremath{\beta}}_{1}=1.$ The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.

DOI
10.1103/physreve.65.016111

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Steady-state properties in the absorbing phase of the 1d1d1d pair contact process model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles \ensuremathρ1,{\ensuremath{\rho}}_{1},\ensuremathρ1​, approaches a stationary value that depends on the annihilation probability (p);(p);(p); the deviation from its ``natural'' value at criticality \ensuremathρ1nat{\ensuremath{\rho}}_{1}^{\mathrm{nat}}\ensuremathρ1nat​ follows a power law: \ensuremathρ1nat\ensuremath−\ensuremathρ1\ensuremath∼(p\ensuremath−pc)\ensuremathβ1{\ensuremath{\rho}}_{1}^{\mathrm{nat}}\ensuremath{-}{\ensuremath{\rho}}_{1}\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}_{1}}\ensuremathρ1nat​\ensuremath−\ensuremathρ1​\ensuremath∼(p\ensuremath−pc​)\ensuremathβ1​ for p>pc.p>{p}_{c}.p>pc​. Monte Carlo simulations yield \ensuremathβ1=0.81.{\ensuremath{\beta}}_{1}=0.81.\ensuremathβ1​=0.81. A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting \ensuremathβ1=1.{\ensuremath{\beta}}_{1}=1.\ensuremathβ1​=1. The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.
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.
I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in 1D yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation universality class.
The steady-state phase diagram of the one-dimensional reaction-diffusion model $2\stackrel{\ensuremath{\rightarrow}}{A}3A,$ $2\stackrel{\ensuremath{\rightarrow}}{A}0$ is studied through the non-Hermitian density matrix renormalization group. In the absence of single-particle diffusion the model reduces to the pair-contact process, which has a phase transition in the universality class of directed percolation (DP) and an infinite number of absorbing steady states. When single-particle diffusion is added, the number of absorbing steady states is reduced to 2 and the model no longer shows DP critical behavior. The exponents $\ensuremath{\theta}={\ensuremath{\nu}}_{\ensuremath{\Vert}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ are calculated numerically. The value of $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ is close to the value of the parity conserving universality class, in spite of the absence of local conservation laws.
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 contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B, B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A, A→A+B, A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension d_{c}=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.