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arXiv:cond-mat/9405029 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Propagation and extinction in branching annihilating random walks

Daniel ben‐Avraham, F. Leyvraz, S. Redner

Published Sep 1, 1994• 48 citations
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Abstract

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.

DOI
10.1103/physreve.50.1843

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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 1993•66 citations
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.
Recently, Takayasu and Tretyakov [Phys. Rev. Lett. 68, 3060 (1992)] studied branching annihilating random walks (BAWs) with n=1--5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd n the models belong to the universality class of directed percolation. For even n the particle number is conserved modulo 2 and the critical behavior is not compatible with directed percolation. In this article I study the BAW with n=4 using time-dependent simulations and finite-size scaling, obtaining precise estimates for various critical exponents. The results are consistent with the conjecture: \ensuremath{\beta}/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=1/2, ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=7/4, \ensuremath{\gamma}=0, \ensuremath{\delta}=2/7, \ensuremath{\eta}=0, and ${\mathrm{\ensuremath{\delta}}}_{\mathit{h}}$=9/2. These critical exponents characterize, respectively, the dependence of the order parameter (\ensuremath{\beta}/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$) and relaxation time (${\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}$/${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$) on system size, the growth of fluctuations (\ensuremath{\gamma}) close to the critical point, the long-time behavior of the probability of survival (\ensuremath{\delta}) and average number of particles (\ensuremath{\eta}) when starting at time zero with just two particles, and finally the decay of the order parameter (${\mathrm{\ensuremath{\delta}}}_{\mathit{h}}$) at the critical point in the presence of an external source.
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.
The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk. It is shown that two-dimensional simulations confirm the field theoretical results with logarithmic corrections. In one dimension, however, if particles exhibit hard core interaction I confirm the very recent predictions of Kwon et al. [Phys. Rev. Lett. 85, 1682 (2000)] that there are two different universality classes depending on the spatial symmetry of the offspring production characterized by beta(S)=0.5 and beta(A)=2. Elaborate analysis of simulation data shows that the order parameter exponent beta does not depend on initial conditions or on diffusion rates of species but strong correction to scaling is observed. By systematic numerical simulations the critical point properties have been explored and initial condition dependence of the dynamical exponents Z and alpha is shown. In the case of a random initial state the particle-density decay at the critical point follows the t(-1/4) law with logarithmic corrections with two offsprings.
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.
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.