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arXiv:cond-mat/9405006 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical exponents for branching annihilating random walks with an even number of offspring

Iwan Jensen

Published Nov 1, 1994• 140 citations
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Abstract

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.

DOI
10.1103/physreve.50.3623

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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}/\ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}\ensuremathν\ensuremath⊥​=1/2, \ensuremathν\ensuremath∥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathν\ensuremath∥​/\ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}\ensuremathν\ensuremath⊥​=7/4, \ensuremath{\gamma}=0, \ensuremath{\delta}=2/7, \ensuremath{\eta}=0, and \ensuremathδh{\mathrm{\ensuremath{\delta}}}_{\mathit{h}}\ensuremathδh​=9/2. These critical exponents characterize, respectively, the dependence of the order parameter (\ensuremath{\beta}/\ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}\ensuremathν\ensuremath⊥​) and relaxation time (\ensuremathν\ensuremath∥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathν\ensuremath∥​/\ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}\ensuremathν\ensuremath⊥​) 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 (\ensuremathδh{\mathrm{\ensuremath{\delta}}}_{\mathit{h}}\ensuremathδh​) at the critical point in the presence of an external source.
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 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.
With extensive Monte Carlo simulations, we present high-precision estimates of the critical exponents of branching annihilating random walks with two offspring, a prototypical model of the directed Ising universality class in one dimension. To estimate the exponents accurately, we propose a systematic method to find corrections to scaling whose leading behavior is supposed to take the form t −χ in the long-time limit at the critical point. Our study shows that χ ≈ 0.75 for the number of particles in defect simulations and χ ≈ 0.5 for other measured quantities, which should be compared with the widely used value of χ = 1. Using χ so obtained, we analyze the effective exponents to find that β/ν ‖ = 0.2872(2), z = 1.7415(5), η = 0.0000(2), and accordingly, β/ν ⊥ = 0.5000(6). Our numerical results for β/ν ‖ and z are clearly different from the conjectured rational numbers β/ν ‖ = $\tfrac{2} {7} $ ≈ 0.2857, z = $\tfrac{7} {4} $ = 1.75 by Jensen [Phys. Rev. E, 50, 3623 (1994)]. Our result for β/ν ⊥, however, is consistent with $\tfrac{1} {2} $ , which is believed to be exact.
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 the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d(c), with d(c)=4 (DP) and d(c)=2 (BARW), we use mean field-theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension d</=d(c), we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field-theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1 and 2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta(1) exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta(1) exponents at the ordinary and special transitions. Many of our predictions are supported using Monte Carlo simulations of two different models belonging to the BARW universality class.
Next to the directed-percolation (DP) universality class, parity-conserving directed-percolation (pcDP; also called parity-conserving branching-annihilating random walks) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent β (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcDP clusters. In particular, we find that all exponents which were close to rationals are even closer, and β=1.000 with the error in the next digit.