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arXiv:cond-mat/0004138 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Systems with superabsorbing states

Pablo I. Hurtado, Miguel A. Muñoz

Published Oct 1, 2000• 2 citations

Abstract

We report on some extensive analyses of a recently proposed model [A. Lipowski, Phys. Rev. E 60, 6255 (1999)] with infinitely many absorbing states. By performing extensive Monte Carlo simulations, we have determined critical exponents and shown strong evidence that this model is not in the directed percolation universality class. The conjecture that this two-dimensional model exhibits a dimensional reduction (behaving as one-dimensional directed percolation) is firmly disproven. The reason for the model not exhibiting standard directed percolation scaling behavior is traced back to the existence of what we call superabsorbing sites, i.e., absorbing sites that cannot be directly activated by the presence of neighboring activity in one or more than one direction. Supporting this claim we present two strong evidences: (i) in one dimension, where superabsorbing sites do not appear at the critical point, the system behaves as directed percolation, and (ii) in a modified two-dimensional variation of the model, defined on a honeycomb lattice, for which superabsorbing sites are very rarely observed, directed percolation behavior is recovered. Finally, a parallel updating version of the model exhibiting a nonequilibrium first-order transition is also reported.

DOI: 10.1103/physreve.62.4633

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We report on some extensive analyses of a recently proposed model [A. Lipowski, Phys. Rev. E 60, 6255 (1999)] with infinitely many absorbing states. By performing extensive Monte Carlo simulations, we have determined critical exponents and shown strong evidence that this model is not in the directed percolation universality class. The conjecture that this two-dimensional model exhibits a dimensional reduction (behaving as one-dimensional directed percolation) is firmly disproven. The reason for the model not exhibiting standard directed percolation scaling behavior is traced back to the existence of what we call superabsorbing sites, i.e., absorbing sites that cannot be directly activated by the presence of neighboring activity in one or more than one direction. Supporting this claim we present two strong evidences: (i) in one dimension, where superabsorbing sites do not appear at the critical point, the system behaves as directed percolation, and (ii) in a modified two-dimensional variation of the model, defined on a honeycomb lattice, for which superabsorbing sites are very rarely observed, directed percolation behavior is recovered. Finally, a parallel updating version of the model exhibiting a nonequilibrium first-order transition is also reported.

At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.

We introduce and solve a model of hardcore particles on a one-dimensional periodic lattice which undergoes an active-absorbing-state phase transition at finite density. In this model, an occupied site is defined to be active if its left neighbor is occupied and the right neighbor is vacant. Particles from such active sites hop stochastically to their right. We show that both the density of active sites and the survival probability vanish as the particle density is decreased below half. The critical exponents and spatial correlations of the model are calculated exactly using the matrix product ansatz. Exact analytical study of several variations of the model reveals that these nonequilibrium phase transitions belong to a new universality class different from the generic active-absorbing-state phase transition, namely, directed percolation.

I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in 1D yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation 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.

We study two nonequilibrium lattice models exhibiting a continuous phase transition from an active state to an absorbing state in which the system is trapped. The models have infinitely many absorbing states. We use one of the models to illustrate how finite-size scaling concepts may be used to enhance computer-simulation studies of the critical behavior. This model is also studied using ordinary steady-state scaling concepts. The results show that the model has the same critical behavior as directed percolation. The applicability of time-dependent simulations, which have proven very efficient in the study of systems with a single absorbing state, is explored extensively using several different initial configurations.