Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour

Grassberger, Peter; Torre, A.

doi:10.1016/0003-4916(79)90207-0

Cited by 563 publications

(617 citation statements)

References 17 publications

Supporting

Mentioning

575

Contrasting

Unclassified

Order By: Relevance

“…To get a good statistics we need to run at least 2.5×10 6 independent samples in all dimensions. From the scaling ansatz for the DP class and similar models [38,39], the physical quantities of interest depend on the relevant parameters → r , t and ∆ = y A − y Ac , only through the scaling variables r 2 t −ζ and ∆t 1/ν || , times some power of r 2 , t or ∆. In the scaling regime, the local fraction of empty sites, averaged over all trials, surviving or not, can be written as…”

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

View full text Add to dashboard Cite

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...

show abstract

Section: Dynamic Monte Carlo Simulationsmentioning

confidence: 99%

Catalysis with competitive reactions: static and dynamical critical behavior

Costa

Figueiredo²

2003

Braz. J. Phys.

View full text Add to dashboard Cite

show abstract

“…Following the formalism developed in Refs. [18,19,20], the state of the system is described by the state vector |P (t) = {σ} P ({σ}, t) |{σ} , whose time evolution is governed by the master equation ∂ t |P (t) =L|P (t) . HereL is the generator of the Markov process that contains the transition rates between the different microstates of the system, |{σ} .…”

mentioning

confidence: 99%

Contact process in heterogeneous and weakly disordered systems

2006

View full text Add to dashboard Cite

The critical behavior of the contact process (CP) in heterogeneous periodic and weakly-disordered environments is investigated using the supercritical series expansion and Monte Carlo (MC) simulations. Phase-separation lines and critical exponents β (from series expansion) and η (from MC simulations) are calculated. A general analytical expression for the locus of critical points is suggested for the weak-disorder limit and confirmed by the series expansion analysis and the MC simulations. Our results for the critical exponents show that the CP in heterogeneous environments remains in the directed percolation (DP) universality class, while for environments with quenched disorder, the data are compatible with the scenario of continuously changing critical exponents.

show abstract

“…Absorbing-state phase transitions arise in the context of spatial stochastic models, and correspond to a transition between an active, fluctuating phase, and an absorbing one, which allows no escape [24,25,27].…”

Section: Introductionmentioning

confidence: 99%

Conserved Sandpile with A Variable Height Restriction

2013

View full text Add to dashboard Cite

We study a restricted-height version of the one-dimensional Oslo sandpile with conserved density, using periodic boundary conditions. Each site has a limiting height which can be either two or three. When a site reaches its limiting height it becomes active and may topple, loosing two particles, which move randomly to nearest-neighbor sites. After a site topples it is randomly assigned a new limiting height. We study the model using meanfield theory and Monte Carlo simulation, focusing on the quasi-stationary state, in which the number of active sites fluctuates about a stationary value. Using finite-size scaling analysis, we determine the critical particle density and associated critical exponents.

show abstract

Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour

Cited by 563 publications

References 17 publications

Catalysis with competitive reactions: static and dynamical critical behavior

Catalysis with competitive reactions: static and dynamical critical behavior

Contact process in heterogeneous and weakly disordered systems

Conserved Sandpile with A Variable Height Restriction

Contact Info

Product

Resources

About