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arXiv:cond-mat/0002214 · Physical Review Letters

Does Hard Core Interaction Change Absorbing-Type Critical Phenomena?

Sungchul Kwon, Jysoo Lee, Hyunggyu Park

Published Aug 21, 2000• 29 citations
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Abstract

It has been generally believed that hard core interaction is irrelevant to absorbing-type critical phenomena because the particle density is so low near an absorbing phase transition. We study the effect of hard core interaction on the $N$-species branching annihilating random walks with two offspring and report that hard core interaction drastically changes the absorbing-type critical phenomena in a nontrivial way. Through a Langevin equation-type approach, we predict analytically the values of the scaling exponents, ${\ensuremath{\nu}}_{\ensuremath{\perp}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, and $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ in one dimension for all $N>1$. Direct numerical simulations confirm our prediction. When the diffusion coefficients for different species are not identical, ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $\ensuremath{\beta}$ vary continuously with the ratios between the coefficients.

DOI
10.1103/physrevlett.85.1682

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It has been generally believed that hard core interaction is irrelevant to absorbing-type critical phenomena because the particle density is so low near an absorbing phase transition. We study the effect of hard core interaction on the NNN-species branching annihilating random walks with two offspring and report that hard core interaction drastically changes the absorbing-type critical phenomena in a nontrivial way. Through a Langevin equation-type approach, we predict analytically the values of the scaling exponents, \ensuremathν\ensuremath⊥=2{\ensuremath{\nu}}_{\ensuremath{\perp}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremathν\ensuremath⊥​=2, z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2, \ensuremathα=1/2\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2\ensuremathα=1/2, and \ensuremathβ=2\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremathβ=2 in one dimension for all N>1N>1N>1. Direct numerical simulations confirm our prediction. When the diffusion coefficients for different species are not identical, \ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremathν\ensuremath⊥​ and \ensuremathβ\ensuremath{\beta}\ensuremathβ vary continuously with the ratios between the coefficients.
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.
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.
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 review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d c , we review the use of mean field theories, whereas in the regime d<d c , where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.