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arXiv:cond-mat/0207720 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonuniversality in the pair contact process with diffusion

Ronald Dickman, Márcio Argollo de Menezes

Published Oct 9, 2002• 36 citations
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Abstract

We study the static and dynamic behavior of the one dimensional pair contact process with diffusion. Several critical exponents are found to vary with the diffusion rate, while the order-parameter moment ratio m=rho(2);/rho;(2) grows logarithmically with system size. The anomalous behavior of m is traced to a violation of scaling in the order parameter probability density, which in turn reflects the presence of two distinct sectors, one purely diffusive, the other reactive, within the active phase. Studies restricted to the reactive sector yield precise estimates for exponents beta and nu( perpendicular ), and confirm finite size scaling. We also determine the value m(c)=1.334 for the parity-conserving universality class in one dimension.

DOI
10.1103/physreve.66.045101

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We study the static and dynamic behavior of the one dimensional pair contact process with diffusion. Several critical exponents are found to vary with the diffusion rate, while the order-parameter moment ratio m=rho(2);/rho;(2) grows logarithmically with system size. The anomalous behavior of m is traced to a violation of scaling in the order parameter probability density, which in turn reflects the presence of two distinct sectors, one purely diffusive, the other reactive, within the active phase. Studies restricted to the reactive sector yield precise estimates for exponents beta and nu( perpendicular ), and confirm finite size scaling. We also determine the value m(c)=1.334 for the parity-conserving universality class in one dimension.
We study the continuous absorbing-state phase transition in the one-dimensional pair contact process with diffusion (PCPD). In previous studies [Dickman and de Menezes, Phys. Rev. E 66, 045101(R) (2002)], the critical point moment ratios of the order parameter showed anomalous behavior, growing with system size rather than taking universal values, as expected. Using the quasistationary simulation method we determine the moments of the order parameter up to fourth order at the critical point, in systems of up to 40,960 sites. Due to strong finite-size effects, the ratios converge only for large system sizes. Moment ratios and associated order-parameter histograms are compared with those of directed percolation. We also report an improved estimate [pc=0.077092(1)] for the location of the critical point in the nondiffusive pair contact process.
We provide finite-size scaling estimates for the dynamical critical exponent of the even parity-conserving universality class of critical behavior through exact numerical diagonalizations of the time evolution operator of an even-parity-conserving contact process. Our data seem to indicate that upon the introduction of a small diffusion rate in the process its critical behavior crosses over to that of the directed percolation universality class. A brief discussion of the many-sector decomposability of parity-conserving contact processes is presented.
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.
Recently it was suggested that a pair contact process with diffusion (PCPD) might represent an independent new universality class different from the directed percolation (DP) and the parity conservation (PC) class. The dynamics in the PCPD are usually controlled by two independent parameters. The critical exponents for the PCPD are known to have different values for varying values of the two independent parameters. However, once the diffusion and annihilation (or coagulation) rate in the PCPD is tuned in a way that the process without offspring production is exactly solvable, a well-defined set of the exponents for the PCPD is obtained. Then dynamics are controlled by only one independent parameter. The obtained critical exponents are different than those of DP and PC. The critical exponents satisfy the generalized hyperscaling relation within numerical errors.
Abstract The question of the universality class of the pair-contact process with diffusion (PCPD) is revisited with an alternative approach. We study persistence in a generalized pair-contact process with diffusion (GPCPD) introduced by Noh and Park (2004 Phys. Rev . E 69 016122). This model allows us to interpolate between directed percolation (DP) and PCPD universality classes. We find that the transition to nonzero persistence is at the same parameter value as the transition to zero number density. We obtain the finite-size scaling and off-critical scaling collapse for persistence and find the critical exponents by fitting phenomenological scaling laws to persistence. While the dynamic scaling exponent z varies continuously in GPCPD, the correlation-time exponent ν ∥ matches with the DP universality class.