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Brazilian Journal of Physics

One-dimensional lattice gas models with infinitely many absorbing states

Carlos E. Fiore, Mário J. de Oliveira

Published Mar 1, 2006• 4 citations
Publisher

Abstract

We determined, by numerical simulations, the critical behavior one-dimensional systems with infinitely many absorbing states. These models are modified versions of the conserved lattice gas model in which two adjacent active particles instead of just one can jump to their neighboring sites. The models studied here are examples of systems with many absorbing states and order parameter coupled to a nondiffusive conserved field that show nontrivial critical behavior.

DOI
10.1590/s0103-97332006000200014

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We determined, by numerical simulations, the critical behavior one-dimensional systems with infinitely many absorbing states. These models are modified versions of the conserved lattice gas model in which two adjacent active particles instead of just one can jump to their neighboring sites. The models studied here are examples of systems with many absorbing states and order parameter coupled to a nondiffusive conserved field that show nontrivial critical behavior.
A one-dimensional conserved lattice-gas model is known to undergo continuous absorbing phase transitions where some of the critical exponents are exactly known. In one dimension, we recently showed that the model is mapped onto a two species reaction A+B→0 with diffusion rate of D_{A}>0 and D_{B}=0. In this work, it is explicitly shown from the scaling theory for A+B→0 that the observed scaling behavior of the conserved lattice-gas model is not associated with the absorbing phase transitions. Instead, the model indeed undergoes a crossover between two different scaling behaviors of A+B→0, the scaling behaviors of equal and unequal initial densities of two species. The crossover is similar to the absorbing transitions in many respects but some important features of continuous transitions such as the diverging fluctuations of an order parameter are absent.
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.
We investigate the critical behavior of a reaction-diffusion system exhibiting a continuous absorbing-state phase transition. The reaction-diffusion system strictly conserves the total density of particles, represented as a nondiffusive conserved field, and allows an infinite number of absorbing configurations. Numerical results show that it belongs to a wide universality class that also includes stochastic sandpile models. We derive microscopically the field theory representing this universality class.
We analyze numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field. The external field is realized as a spontaneous creation of active particles that drives the system away from criticality. Nevertheless, the order parameter obeys certain scaling laws for sufficiently small external fields. These scaling laws are investigated and the corresponding exponents are determined in various dimensions (D=2,3,4,5). At the so-called upper critical dimension D(c)=4 one has to modify the usual scaling laws by logarithmic corrections.
We analyze numerically the critical behavior of a conserved lattice gas that was recently introduced as an example of the new universality class of absorbing phase transitions with a conserved field [Phys. Rev. Lett. 85, 1803 (2000)]. We determine the critical exponent of the order parameter as well as the critical exponent of the order parameter fluctuations in D=2,3,4,5 dimensions. A comparison of our results and those obtained from a mean-field approach and a field theory suggests that the upper critical dimension of the absorbing phase transition is four.
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.