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Modern Physics Letters B

IRREVERSIBLE TRANSITIONS IN A TWO SPECIES BRANCHING ANNIHILATING RANDOM WALKER PROCESS

M. Hoyuelos, Ezequiel Vicente Albano, Héctor O. Mártin

Published Nov 10, 1996• 0 citations
Publisher

Abstract

We introduce a branching annihilating random walker process with two species, particles A and B, which diffuse creating particles of opposite kind (A→A+B, B→B+A) and annihilating instantaneously (A+B→0) when they meet. The model is defined in a one dimensional discrete lattice. For particles A and B, the rate of jumping are pA and pB, and the rate of branching (1−pA) and (1−pB), respectively (0≤pA, pB≤1). In the [pA, pB]-plane it is found two different phases: the vacuum state and the active phase with finite density of particles. The system undergoes irreversible second order phase transitions between these states along a critical line. Monte Carlo results show that the transitions belong to the same universality class as directed percolation.

DOI
10.1142/s0217984996001450

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We study the effects of hard core (HC) interactions between different species of particles on two-species branching annihilating random walks with one offspring(BAW2_22​(1)). The single-species model belongs to the directed percolation (DP) universality class. In the BAW2_22​(1) model, a particle creates one particle of the same species in its neighborhood with the probability σ(1−p)σ(1-p)σ(1−p) and of the different species with (1−σ)(1−p)(1-σ)(1-p)(1−σ)(1−p), where ppp is the hopping probability. Without HC interactions, this model always exhibits the DP-type absorbing transition for all σσσ. Even with HC interactions, the nature of the phase transitions does not change except near σ=0σ=0σ=0, where the HC interaction destabilizes and completely wipes away the absorbing phase. The model is always active except at the annihilation fixed point of zero branching rate (p=1p=1p=1). Critical behavior near the annihilation fixed point is characterized by exponents β=ν⊥=1/2β= ν_\perp =1/2β=ν⊥​=1/2 and ν∣∣=1ν_{||} = 1ν∣∣​=1.
We present some exact results on the behavior of branching and annihilating random walks, both in the directed percolation and parity conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the nontrivial pure annihilation (PA) model, whose correlation and response function we compute exactly. With this, the nonuniversal threshold value for having a phase transition in the simplest system belonging to the directed percolation universality class is found to coincide with previous nonperturbative renormalization group (RG) approximate results. We also show that the parity conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
Using Monte Carlo simulations we have studied the transition from an ``active'' steady state to an absorbing ``inactive'' state for two versions of the branching annihilating random walks with parity conservation on a square lattice. In the first model the randomly walking particles annihilate when they meet and the branching process creates two additional particles; in the second case we distinguish particles and antiparticles created and annihilated in pairs. Quite distinct critical behavior is found in the two cases, raising the question of what determines universality in these kinds of systems.
In this work, we investigate the critical behavior of a model describing the parity-conserving branching and annihilating process of random walkers. The model is simulated on a one dimensional lattice on which the sites can be occupied by multiple particles with a finite annhilation probability. We determine the threshold of the phase transition between the statistically stationary active state and the absorbing state. From steady-state simulations and a finite-size scaling analysis, we calculate the order-parameter, order-parameter fluctuations, and spacial correlation length critical exponents. Further, we follow the short-time critical relaxation to provide a set of relevant dynamical critical exponents. We check the validity of the hyperscaling relation for both sets of stationary and dynamical critical exponents. These are consistent with the BARW-PC universality class.
We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias ε. For ε=0, three universality classes-directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)-are typically observed in such systems. We find that the introduction of ε does not change the DP universality class but alters the other two universality classes. For nonzero ε, the PCPD class crosses over to DP, and the PC class changes to a new universality class.
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.