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Physical Review A

Scaling analysis of time-dependent simulations of the three-dimensional contact process

Iwan Jensen

Published Dec 1, 1992• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.46.7393

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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.
I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. I study the model in the vicinity of the critical point, and in the subcritical region, via time-dependent Monte Carlo simulations. The method used in the subcritical region is very efficient. The results for the directly measured critical exponents, \ensuremath{\nu}=1.11\ifmmode\pm\else\textpm\fi{}0.01, \ensuremath{\eta}=0.114\ifmmode\pm\else\textpm\fi{}0.004, and z=1.052\ifmmode\pm\else\textpm\fi{}0.003, are consistent with those of directed percolation. \ensuremath{\delta}=0.732\ifmmode\pm\else\textpm\fi{}0.004 is obtained from the hyperscaling relation 4\ensuremath{\delta}+2\ensuremath{\eta}=dz, and \ensuremath{\beta}=0.813\ifmmode\pm\else\textpm\fi{}0.011 from \ensuremath{\beta}=\ensuremath{\nu}\ensuremath{\delta}. These results are the most precise so far for a three-dimensional model with directed percolation critical behavior.
The absorbing-state transition in the three-dimensional contact process with and without quenched randomness is investigated by means of Monte Carlo simulations. In the clean case, a reweighting technique is combined with a careful extrapolation of the data to infinite time to determine with high accuracy the critical behavior in the three-dimensional directed percolation universality class. In the presence of quenched spatial disorder, our data demonstrate that the absorbing-state transition is governed by an unconventional infinite-randomness critical point featuring activated dynamical scaling. The critical behavior of this transition does not depend on the disorder strength, i.e., it is universal. Close to the disordered critical point, the dynamics is characterized by the nonuniversal power laws typical of a Griffiths phase. We compare our findings to the results of other numerical methods, and we relate them to a general classification of phase transitions in disordered systems based on the rare region dimensionality.
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.
We study the two-dimensional contact process (CP) with quenched disorder (DCP), and determine the static critical exponents $\ensuremath{\beta}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}.$ The dynamic behavior is incompatible with scaling, as applied to models (such as the pure CP) that have a continuous phase transition to an absorbing state. We find that the survival probability (starting with all sites occupied), for a finite-size system at the critical point, decays according to a power law, as does the off-critical density autocorrelation function. Thus the critical exponent ${\ensuremath{\nu}}_{||},$ which governs the relaxation time, is undefined, since the characteristic relaxation time is itself undefined. The logarithmic time dependence found in recent simulations of the critical DCP [A. G. Moreira and R. Dickman, Phys. Rev. E 54, R3090 (1996)] is further evidence of violation of scaling. A simple argument based on percolation cluster statistics yields a similar logarithmic evolution.
We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to 10;{10} and system sizes up to 8000x8000 sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.