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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical behavior of a two-species reaction-diffusion problem

J. E. de Freitas, LIACIR S. LUCENA, L. R. da Silva, H. J. Hilhorst

Published Jun 1, 2000• 31 citations
Publisher

Abstract

We present a Monte Carlo study in dimension $d=1$ of the two-species reaction-diffusion process $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A.$ Below a critical value ${\ensuremath{\rho}}_{c}$ of the conserved total density $\ensuremath{\rho}$ the system falls into an absorbing state without B particles. Above ${\ensuremath{\rho}}_{c}$ the steady state B particle density ${\ensuremath{\rho}}_{B}^{\mathrm{st}}$ is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities $\ensuremath{\rho}$ between zero and some maximum density. From finite-size scaling we obtain the steady state exponents $\ensuremath{\beta}=0.435(10),\ensuremath{\nu}=2.21(5),$ and $\ensuremath{\eta}=\ensuremath{-}0.606(4)$ for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value ${\ensuremath{\theta}}^{\ensuremath{'}}=0.30(2),$ in agreement with the theoretical relation ${\ensuremath{\theta}}^{\ensuremath{'}}=\ensuremath{-}\ensuremath{\eta}/2$ due to Van Wijland et al. [Physica A 251, 179 (1998)].

DOI
10.1103/physreve.61.6330

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In a recent paper, de Freitas et al. [Phys. Rev. E 61, 6330 (2000)] presented simulational results for the critical exponents of the two-species reaction-diffusion system $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A$ in dimension $d=1.$ In particular, the correlation length exponent was found as $\ensuremath{\nu}=2.21(5)$ in contradiction to the well-known relation $\ensuremath{\nu}=2/d.$ In this Comment, the symmetry arguments leading to exact critical exponents for the universality class to which this reaction-diffusion system belongs are concisely reconsidered.

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We present a Monte Carlo study in dimension d=1d=1d=1 of the two-species reaction-diffusion process A+B\ensuremath→2BA+\stackrel{\ensuremath{\rightarrow}}{B}2BA+B\ensuremath→​2B and B\ensuremath→A.\stackrel{\ensuremath{\rightarrow}}{B}A.B\ensuremath→​A. Below a critical value \ensuremathρc{\ensuremath{\rho}}_{c}\ensuremathρc​ of the conserved total density \ensuremathρ\ensuremath{\rho}\ensuremathρ the system falls into an absorbing state without B particles. Above \ensuremathρc{\ensuremath{\rho}}_{c}\ensuremathρc​ the steady state B particle density \ensuremathρBst{\ensuremath{\rho}}_{B}^{\mathrm{st}}\ensuremathρBst​ is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities \ensuremathρ\ensuremath{\rho}\ensuremathρ between zero and some maximum density. From finite-size scaling we obtain the steady state exponents \ensuremathβ=0.435(10),\ensuremathν=2.21(5),\ensuremath{\beta}=0.435(10),\ensuremath{\nu}=2.21(5),\ensuremathβ=0.435(10),\ensuremathν=2.21(5), and \ensuremathη=\ensuremath−0.606(4)\ensuremath{\eta}=\ensuremath{-}0.606(4)\ensuremathη=\ensuremath−0.606(4) for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value \ensuremathθ\ensuremath′=0.30(2),{\ensuremath{\theta}}^{\ensuremath{'}}=0.30(2),\ensuremathθ\ensuremath′=0.30(2), in agreement with the theoretical relation \ensuremathθ\ensuremath′=\ensuremath−\ensuremathη/2{\ensuremath{\theta}}^{\ensuremath{'}}=\ensuremath{-}\ensuremath{\eta}/2\ensuremathθ\ensuremath′=\ensuremath−\ensuremathη/2 due to Van Wijland et al. [Physica A 251, 179 (1998)].
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension dc=4{d}_{c}=4dc​=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes $\stackrel{\ensuremath{\rightarrow}}{A}A+A,$ $A+\stackrel{\ensuremath{\rightarrow}}{A}A$, and $\stackrel{\ensuremath{\rightarrow}}{A}0$. We study a hierarchy of such DP processes for particle species A,B,…A,B,\dots{}A,B,…, unidirectionally coupled via the reactions $\stackrel{\ensuremath{\rightarrow}}{A}B,\dots{}$ (with rates ${\ensuremath{\mu}}_{\mathrm{AB}},\dots{})$. When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents ${\ensuremath{\beta}}_{i}$ which are markedly reduced at each hierarchy level i> 2i>~2i> 2. This scenario can be understood on the basis of the mean-field rate equations, which yield ${\ensuremath{\beta}}_{i}{=1/2}^{i\ensuremath{-}1}$ at the multicritical point. Using field-theoretic renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we identify a new crossover exponent $\ensuremath{\varphi}$, and compute $\ensuremath{\varphi}=1+O({\ensuremath{\epsilon}}^{2})$ in the multicritical regime (for small ${\ensuremath{\mu}}_{\mathrm{AB}})$ of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, ${\ensuremath{\beta}}_{2}=1/2\ensuremath{-}\ensuremath{\epsilon}/8+O({\ensuremath{\epsilon}}^{2})$. Outside the multicritial region, we discuss the crossover to ordinary DP behavior, with the density exponent ${\ensuremath{\beta}}_{1}=1\ensuremath{-}\ensuremath{\epsilon}/6+O({\ensuremath{\epsilon}}^{2})$. Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< 3d<~3d< 3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.
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.
In a recent paper, de Freitas et al. [Phys. Rev. E 61, 6330 (2000)] presented simulational results for the critical exponents of the two-species reaction-diffusion system $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A$ in dimension d=1.d=1.d=1. In particular, the correlation length exponent was found as $\ensuremath{\nu}=2.21(5)$ in contradiction to the well-known relation $\ensuremath{\nu}=2/d.$ In this Comment, the symmetry arguments leading to exact critical exponents for the universality class to which this reaction-diffusion system belongs are concisely reconsidered.
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.
Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks with n=1-5 offsprings. These models exhibit a continuous phase transition to an absorbing state. Steady-state simulations yielded an estimate for the order parameter critical exponent \ensuremath{\beta} different from that of directed percolation. This result is quite surprising, as the universality class of directed percolation is known to be very robust. I have studied the critical behavior of the one-dimensional model with n=1 and 3 using time-dependent Monte Carlo simulations, and determined three critical exponents, all of which are in agreement with directed percolation.
We study the dynamic scaling behavior of a monomer-dimer model with repulsive itneractions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. This transition does not belong to the conventional directed percolation universality class. The values of dynamic scaling exponents are estimated by Monte Carlo simulations for two distinct initial configurations, one near an absorbing state and the other with an interface between two different absorbing states. We confirm that the critical behavior is consistent with that of the models with the mass conservation of modulo 2.