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Generalized scaling for models with multiple absorbing states

J. F. F. Mendes, Ronald Dickman, Malte Henkel, M. Ceu Marques

Published Jan 1, 1994• 120 citations
OpenAlex

Abstract

At a continuous transition into a nonunique absorbing state, particle systems may exhibit nonuniversal critical behavior, in apparent violation of hyperscaling. We propose a generalized scaling theory for dynamic critical behavior at a transition into an absorbing state, which is capable of describing exponents which vary according to the initial configuration. The resulting hyperscaling relation is supported by simulations of two lattice models.

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At a continuous transition into a nonunique absorbing state, particle systems may exhibit nonuniversal critical behavior, in apparent violation of hyperscaling. We propose a generalized scaling theory for dynamic critical behavior at a transition into an absorbing state, which is capable of describing exponents which vary according to the initial configuration. The resulting hyperscaling relation is supported by simulations of two lattice models.
We present a general scaling theory for the surface critical behavior of nonequilibrium systems with phase transitions into absorbing states. The theory allows for two independent surface exponents which satisfy generalized hyperscaling relations. As an application we study a generalized version of directed percolation with two absorbing states. We find two distinct surface universality classes associated with inactive and reflective walls. Our results indicate that the exponents associated with these two surface universality classes are closely connected.
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.
We consider two different systems exhibiting a continuous phase transition into an absorbing state. Both models belong to the same universality class; i.e., they are characterized by the same scaling functions and the same critical exponents. Varying the range of interactions, we examine the crossover from the mean-field-like to the non-mean-field scaling behavior. A phenomenological scaling form is applied in order to describe the full crossover region, which spans several decades. Our results strongly support the hypothesis that the crossover function is universal.
One-dimensional systems with an infinite number of absorbing states exhibit a phase transition that is not fully understood yet. Their static critical exponents are universal and belong in the Reggeon field theory (or directed percolation) universality class. However, exponents associated with the spreading of a localized seed appear to be nonuniversal depending on the nature of the initial condition. We investigate this problem by integrating numerically a non-Markovian Langevin equation proposed recently to describe such phase transitions. We find that the static critical exponents are universal, as expected. On the other hand, the Langevin equation reproduces the nonuniversal behavior observed in microscopic models for exponents associated with the spreading of an initially localized seed and satisfies the generalized hyperscaling relation proposed for those systems.
In this work, we investigate the critical behavior of a model describing the parity-conserving branching and annihilating process of random walkers. The model is simulated on a one dimensional lattice on which the sites can be occupied by multiple particles with a finite annhilation probability. 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 spacial correlation length critical exponents. Further, we follow the short-time critical relaxation to provide a set of relevant dynamical critical exponents. We check the validity of the hyperscaling relation for both sets of stationary and dynamical critical exponents. These are consistent with the BARW-PC universality class.