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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium phase transitions in systems with infinitely many absorbing states

Iwan Jensen, Ronald Dickman

Published Sep 1, 1993• 161 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.48.1710

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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.
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.
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, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.
I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. Time-dependent Monte Carlo simulations in the vicinity of the critical point yield very accurate estimates for the location of the critical point and several critical exponents. Time-dependent simulations in the supercritical and subcritical regimes yield estimates for further critical exponents from the asymptotic behavior of the survival probability and the average number of particles. The results in the super- and subcritical regimes are also analyzed by utilizing the scaling laws describing the behavior of the model. The results for the critical exponents are the best obtained so far for a three-dimensional model with directed-percolation critical behavior.
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.