Critical exponents for branching annihilating random walks with an even number of offspring

Jensen, Iwan

doi:10.1103/physreve.50.3623

Cited by 149 publications

(217 citation statements)

References 40 publications

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“…Consider, for instance, Eq. (31). By changing the scale of the time variable by an integer m we can write, at the critical point, P (mt) = m −δ P (t).…”

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

“…Another way of studying the dynamical critical behavior of the model is by employing an epidemic analysis [29,30,31]. We measured the survival probability P (t), the number of empty sites n v (t), and the mean square displacement from the origin R 2 (t), from an initial state containing only a single empty site at the center of the lattice.…”

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

“…Thus, the configuration of the system in the beginning of the simulations is very close to the absorbing state. That study was made by employing an epidemic analysis [29][30][31]. The dynamical critical exponents of the model are the same as those of Directed Percolation.…”

Section: Introductionmentioning

confidence: 99%

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Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

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We studied in this work a competitive reaction model between monomers on a catalyst. The catalyst is represented by hypercubic lattices in d = 1, 2 and 3 dimensions. The model is described by the following reactions: A + A → A2 and A + B → AB, where A and B are two monomers that arrive at the surface with probabilities y A and y B , respectively. The model is studied in the adsorption controlled limit where the reaction rate is infinitely larger than the adsorption rate. We employ site and pair mean-field approximations as well as static and dynamical Monte Carlo simulations. We show that, for all d, the model exhibits a continuous phase transition between an active steady state and a B-absorbing state, when the parameter y A is varied through a critical value. Monte Carlo simulations and finite-size scaling analysis near the critical point are used to determine the static critical exponents β, ν ⊥ and the dynamical critical exponents ν || , δ, η and z. The results found for this competitive reaction model are in accordance with the conjecture of Grassberger, which states that any system undergoing a continuous phase transition from an active steady state to a single absorbing state, exhibits the same critical behavior of the directed percolation universality class. I IntroductionIn the course of the last decade the statistical mechanics community has made great progress in the study of nonequilibrium phenomena. Until now, we do not have a complete theory accounting for the nonequilibrium systems. The fundamental concept of a Gibbsian distribution of states in equilibrium has no counterpart in the nonequilibrium situation. This happens because many of these systems do not present even an hamiltonian function and, if it is possible to define an hamiltonian, the detailed balance would be violated.Examples of recent problems on nonequilibrium processes include markets [1, 2], rain precipitation [3], sandpiles [4] and conserved contact process [5]. There is also a great interest in modeling interface growth [6,7], traffic flow [8], temperature dependent catalytic reactions [9], etc. Nonequilibrium magnetic systems, with a well defined hamiltonian, have been also studied in the context of nonequilibrium processes [10,11] as well.For the equilibrium systems we can induce phase transitions by changing some external parameters. Usually, the temperature is the selected control parameter to study phase transitions between equilibrium states. In the case of continuous phase transitions, at the critical point, long range correlations are established inside the system and a set of critical exponents can be defined to describe the critical behavior of some thermodynamic properties. The renormalization group theory [12] is a well known theory that allows the calculation of these critical exponents.We can also consider external constraints for the nonequilibrium systems that can drive the dynamical behavior of the system. The nature of the external parameter depends on the nature of the system. For instance, in an epidemic mode...

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“…Consider, for instance, Eq. (31). By changing the scale of the time variable by an integer m we can write, at the critical point, P (mt) = m −δ P (t).…”

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

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Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

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“…In terms of the domain wall, there is a single absorbing state (vacuum) with parity conservation in the number of domain walls, since spin flips change it only in pairs. Other examples in the DI class includes the interacting monomer-dimer model [21], the branching-annihilating random walks with an even number of offspring [12,22], and generalized versions of the contact process [5,23]. There also exist models in the DI universality class that have infinitely many absorbing states [24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Universality class of absorbing transitions with continuously varying critical exponents

Noh

Park

2004

Phys. Rev. E

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The well-established universality classes of absorbing critical phenomena are directed percolation (DP) and directed Ising (DI) classes. Recently, the pair contact process with diffusion (PCPD) has been investigated extensively and claimed to exhibit a new type of critical phenomena distinct from both DP and DI classes. Noticing that the PCPD possesses a long-term memory effect, we introduce a generalized version of the PCPD (GPCPD) with a parameter controlling the memory effect. The GPCPD connects the DP fixed point to the PCPD point continuously. Monte Carlo simulations show that the GPCPD displays novel type critical phenomena which are characterized by continuously varying critical exponents. The same critical behaviors are also observed in models where two species of particles are coupled cyclically. We suggest that the long-term memory may serve as a marginal perturbation to the ordinary DP fixed point.

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“…An absorbing transition is observed for all n with finite annihilation rates α. In 1 + 1 dimensions, the best known estimates of critical exponents for the PC class are β T = 0.92 ± 0.02, ν = 3.22 ± 0.06 and ν ⊥ = 1.83 ± 0.03 [34].…”

Section: B Phase Transitions In Systems With Absorbing Statesmentioning

confidence: 99%

Depinning transitions in interface growth models

Reis

2003

Braz. J. Phys.

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Pinning-depinning transitions are roughening transitions separating a growing phase and pinned (or blocked) one, and are frequently connected to transitions into absorbing states. In this review, we discuss lattice growth models exhibiting this type of dynamic transition. Driven growth in media with impurities, the competition between deposition and desorption and deposition of poisoning species are some of the physical mechanisms responsible for the transitions, leading to different types of stochastic growth rules. The growth models are classified according to the those mechanisms and possible applications are shown, which include suggestions of experimental realizations of directed percolation transitions. I IntroductionThe study of surface and interface growth processes attracts much interest from the technological point of view mainly because it helps to understand the growth mechanisms of nanostructures and, consequently, their physical properties [1]. From the theoretical point of view, they motivated significant advances in non-equilibrium Statistical Mechanics and related fields [2,3]. Theoretical modeling is usually based on stochastic differential equations or on discrete atomistic models. In various models, changing one parameter leads to a roughening transition between a rough and a smooth phase. Two well known examples are the transition in the Kardar-Parisi-Zhang equation in dimensions d ≥ 3, separating regimes with linear (smooth) and nonlinear growth, and temperature-induced roughening transitions in equilibrium conditions [2]. On the other hand, in the so-called depinning transitions, the interface propagates only in the rough phase, being pinned in the smooth phase. Several mechanisms may be responsible for this type of transition, such as the presence of impurities in the growth media, competition between deposition and evaporation or formation of poisoning species at the growing surface. One of the interesting features of depinning transitions is that they frequently can be mapped onto transitions into absorbing states, whose most prominent example is directed percolation (DP). Different connections of depinning transitions to DP are found in discrete and continuous growth models, as well as connections with other classes of transitions to absorbing states [4] and, eventually, with statistical equilibrium transitions. Besides the fundamental interest of these systems, the large variety of interface growth phenomena observed in nature suggests them as strong candidates to experimental realizations of models of dynamic transitions, such as DP (see e. g. Ref.[5]).The aim of the present work is to review the basic features of lattice growth models exhibiting depinning transitions. Before introducing these systems, we will briefly summarize the theory and phenomenology concerning interface growth and phase transitions into absorbing states (Sec. II). Then we will discuss the problem of growing interfaces in disordered media with quenched disorder, in which transitions in the DP class or in the cl...

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Critical exponents for branching annihilating random walks with an even number of offspring

Cited by 149 publications

References 40 publications

Catalysis with competitive reactions: static and dynamical critical behavior

Catalysis with competitive reactions: static and dynamical critical behavior

Universality class of absorbing transitions with continuously varying critical exponents

Depinning transitions in interface growth models

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