Catalysis with competitive reactions: static and dynamical critical behavior

Abstract

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...

A simple model for two competing chemical reactions

“…This result is independent of a and is exactly the same we have obtained in our previous work [4]. This is not a surprise since in the site approximation the probability of a reaction to occur does not depend on the presence of other particles.…”

“…We found that the ratio b/m ? is (a) 0.251 ( (2) and (c) 0.264 (4). Based on these results we can conclude that…”

“…AB, where A and B are two monomers that arrive at the surface with probabilities y A and y B , respectively. We studied the model in one, two and three spatial dimensions [2][3][4]. The model presents a continuous phase transition from a reactive steady state into a B-absorbing state, and our results showed that this phase transition belongs to the same universality class of the Directed Percolation (DP).…”

Moment ratios and dynamic critical behavior of a reactive system with several absorbing configurations

“…From Table II it is clear that the exponents we have obtained follow the relation given by equation (18). We have also considered an arbitrary initial configuration that is different from the natural configuration of the system.…”

“…In particular, we are interested in the study of catalytic reaction models with competitive reactions between two types of monomers: the autocatalytic reaction A + A → A 2 [15] and the simple monomer-monomer reaction A + B → AB [16]. This competitive reaction model was studied, at zero temperature, through the dynamical mean-field approximation and Monte Carlo simulations [17,18]. The results showed that the critical exponents of the model are in the DP universality class, and that the absorbing state is unique, with the lattice completely poisoned by monomers of type B.…”