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arXiv:cond-mat/9812005 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Branching annihilating random walks with parity conservation on a square lattice

György Szabó, Maria Augusta Santos

Published Mar 1, 1999• 8 citations
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Abstract

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.

DOI
10.1103/physreve.59.r2509

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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 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.
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.
In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. 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 special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.
We study a one-dimensional, nonequilibrium Potts-like model that has q symmetric absorbing states. For q=2, as expected, the model belongs to the parity-conserving universality class. For q=3 the critical behavior depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity-conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behavior.
We investigate the temporal evolution and spatial propagation of branching annihilating random walks (BAWs) in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite density wave, or an extinction may occur, in which the number of particles vanishes in the long-time limit. The number parity conserving case where two offspring are produced in each branching event can be solved exactly for a unit reaction probability, from which qualitative features of the transition between propagation and extinction, as well as intriguing parity-specific effects, are elucidated. An approximate analysis is developed to treat this transition for general BAW processes. A scaling description suggests that the critical exponents that describe the vanishing of the particle density at the transition are unrelated to those of conventional models, such as Reggeon field theory.