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arXiv:cond-mat/0106102 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Generalized contact process with n absorbing states

Jef Hooyberghs, Enrico Carlon, Carlo Vanderzande

Published Aug 30, 2001• 18 citations

Abstract

We investigate the critical properties of a one-dimensional stochastic lattice model with n (permutation symmetric) absorbing states. We analyze the cases with n</=4 by means of the nonhermitian density-matrix renormalization group. For n=1 and n=2 we find that the model is, respectively, in the directed percolation and parity conserving universality class, consistent with previous studies. For n=3 and n=4, the model is in the active phase in the whole parameter space and the critical point is shifted to the limit of one infinite reaction rate. We show that in this limit, the dynamics of the model can be mapped onto that of a zero temperature n-state Potts model. On the basis of our numerical and analytical results, we conjecture that the model is in the same universality class for all n>/=3 with exponents z=nu( ||)/nu( perpendicular)=2, nu( perpendicular)=1, and beta=1. These exponents coincide with those of the multispecies (bosonic) branching annihilating random walks. For n=3 we also show that, upon breaking the symmetry to a lower one (Z2), one gets a transition either in the directed percolation, or in the parity conserving class, depending on the choice of parameters.

DOI: 10.1103/physreve.64.036124

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We investigate the critical properties of a one-dimensional stochastic lattice model with n (permutation symmetric) absorbing states. We analyze the cases with n</=4 by means of the nonhermitian density-matrix renormalization group. For n=1 and n=2 we find that the model is, respectively, in the directed percolation and parity conserving universality class, consistent with previous studies. For n=3 and n=4, the model is in the active phase in the whole parameter space and the critical point is shifted to the limit of one infinite reaction rate. We show that in this limit, the dynamics of the model can be mapped onto that of a zero temperature n-state Potts model. On the basis of our numerical and analytical results, we conjecture that the model is in the same universality class for all n>/=3 with exponents z=nu( ||)/nu( perpendicular)=2, nu( perpendicular)=1, and beta=1. These exponents coincide with those of the multispecies (bosonic) branching annihilating random walks. For n=3 we also show that, upon breaking the symmetry to a lower one (Z2), one gets a transition either in the directed percolation, or in the parity conserving class, depending on the choice of parameters.

The steady-state phase diagram of the one-dimensional reaction-diffusion model $2\stackrel{\ensuremath{\rightarrow}}{A}3A,$ $2\stackrel{\ensuremath{\rightarrow}}{A}0$ is studied through the non-Hermitian density matrix renormalization group. In the absence of single-particle diffusion the model reduces to the pair-contact process, which has a phase transition in the universality class of directed percolation (DP) and an infinite number of absorbing steady states. When single-particle diffusion is added, the number of absorbing steady states is reduced to 2 and the model no longer shows DP critical behavior. The exponents $\ensuremath{\theta}={\ensuremath{\nu}}_{\ensuremath{\Vert}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ are calculated numerically. The value of $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ is close to the value of the parity conserving universality class, in spite of the absence of local conservation laws.

We study a one-dimensional, nonequilibrium Potts-like model that has q symmetric absorbing states. For q=2, as expected, the model belongs to the parity-conserving universality class. For q=3 the critical behavior depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity-conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behavior.

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.

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.

Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.

In this dissertation we address the problem of second order phase transitions with absorbing states. We analyze the critical behavior of a one dimensional model simulated on one dimensional chains. The sites on the linear chain can be multiple occupied by particles that annhilates each other with a finite probability. It also presents parity-conserving branching with an even number of offsprings. The number of particles are constant modulo 2. 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 special correlation length critical exponents. We also followed the short-time critical relaxation to determine some dynamical critical exponents. The hyperscaling relation is checked for both sets of stationary and dynamical critical exponents which we found to be consistent with the BARW-PC universality class.