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arXiv:cond-mat/0701315

The double domain structure of pair contact process with diffusion

Sungchul Kwon, Yup Kim

Published Jan 15, 2007• 0 citations
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Abstract

We investigate the domain structure of pair contact process with diffusion (PCPD). PCPD is a stochastic reaction-diffusion model which evolves by the competition of two binary reactions, $2A \to 3A$ and $2A \to 0$. In addition, each particle diffuses isotropically, which leads to the bidirectional coupling between solitary particles and pairs. The coupling from pairs to solitary particles is linear, while the opposite coupling is quadratic. The spreading domain formed from localized activities in vacuum consists of two regions, the coupled region of size $R_p$ where pairs and solitary particles coexist and the uncoupled region of size $R_U$ where only solitary particles exist respectively. As the size of the whole domain $R$ is given as $R=R_p + R_U$, $R_p$ and $R_U$ are the basic length scales of PCPD. At criticality, $R_p$ and $R_U$ scale as $R_p \sim t^{1/Z_p}$ and $R_U \sim t^{1/Z_U}$ with $Z_U > Z_p$. We estimate $Z_p =1.61(1)$ and $Z_U =1.768(8)$. Hence, the correction to the scaling of $R$, $Q=R_U /R_p$ extremely slowly decays, which makes it practically impossible to identify the asymptotic scaling behavior of $R$. In addition to the generic feature of the bidirectional coupling, the double domain structure is another reason for the extremely slow approach to the asymptotic scaling regime of PCPD.

DOI
10.48550/arxiv.cond-mat/0701315

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We investigate the domain structure of pair contact process with diffusion (PCPD). PCPD is a stochastic reaction-diffusion model which evolves by the competition of two binary reactions, 2A→3A2A \to 3A2A→3A and 2A→02A \to 02A→0. In addition, each particle diffuses isotropically, which leads to the bidirectional coupling between solitary particles and pairs. The coupling from pairs to solitary particles is linear, while the opposite coupling is quadratic. The spreading domain formed from localized activities in vacuum consists of two regions, the coupled region of size RpR_pRp​ where pairs and solitary particles coexist and the uncoupled region of size RUR_URU​ where only solitary particles exist respectively. As the size of the whole domain RRR is given as R=Rp+RUR=R_p + R_UR=Rp​+RU​, RpR_pRp​ and RUR_URU​ are the basic length scales of PCPD. At criticality, RpR_pRp​ and RUR_URU​ scale as Rp∼t1/ZpR_p \sim t^{1/Z_p}Rp​∼t1/Zp​ and RU∼t1/ZUR_U \sim t^{1/Z_U}RU​∼t1/ZU​ with ZU>ZpZ_U > Z_pZU​>Zp​. We estimate Zp=1.61(1)Z_p =1.61(1)Zp​=1.61(1) and ZU=1.768(8)Z_U =1.768(8)ZU​=1.768(8). Hence, the correction to the scaling of RRR, Q=RU/RpQ=R_U /R_pQ=RU​/Rp​ extremely slowly decays, which makes it practically impossible to identify the asymptotic scaling behavior of RRR. In addition to the generic feature of the bidirectional coupling, the double domain structure is another reason for the extremely slow approach to the asymptotic scaling regime of PCPD.
We investigate the domain structure of the pair contact process with diffusion (PCPD). PCPD is a stochastic reaction-diffusion model which evolves by the competition of two binary reactions, 2A-->3A and 2A-->0 . In addition, each particle diffuses isotropically, which leads to the bidirectional coupling between solitary particles and pairs. The spreading domain formed from localized activities in vacuum consists of two regions: The coupled region of size R{p} where pairs and solitary particles coexist and the uncoupled region of size R{U} where only solitary particles exist, respectively. The size of the whole domain R is given as R=R{p}+R{U} . At criticality, R{p} and R{U} scale as R{p} approximately t(1/Z{p}) and R{U} approximately t(1/Z{U}) with Z{U}>Z{p}. We estimate Z{p}=1.61(1) and Z{U}=1.768(8) . Hence the correction to the scaling of R,Q=R{U}/R{p} slowly decays extremely, which makes it practically impossible to identify the asymptotic scaling behavior of R. The double domain structure is another reason for the extremely slow approach to the asymptotic scaling regime of PCPD.
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.
We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent δ. To obtain an accurate estimate of δ, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 62, 469 (2013)KPSJAS0374-488410.3938/jkps.62.469]. For small diffusion rate (d≤0.5), the leading corrections-to-scaling term is found to be ∼t^{-0.15}, whereas for large diffusion rate (d=0.95) it is found to be ∼t^{-0.5}. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent δ is 0.173(3) irrespective of d. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At d=0, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent ϕ=2.6(1). We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At d=1, the crossover from the mean field PCPD to the PCPD is described by ϕ=2 which is argued to be exact.
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.
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.
Crossover behaviors from the pair contact process with diffusion (PCPD) and the driven PCPD (DPCPD) to the directed percolation (DP) are studied in one dimension by introducing a single particle annihilation and/or branching dynamics. The crossover exponents phi are estimated numerically as 1/phi approximately 0.58 +/- 0.03 for the PCPD and 1/phi approximately 0.49+/-0.02 for the DPCPD. Nontriviality of the PCPD crossover exponent strongly supports the non-DP nature of the PCPD critical scaling, which is further evidenced by the anomalous critical amplitude scaling near the PCPD point. In addition, we find that the DPCPD crossover is consistent with the mean field prediction of the tricritical DP class as expected.