© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/9704035 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Mean-field analysis and Monte Carlo study of an interacting two-species reaction model

Kevin Brown, Kevin E. Bassler, D. A. Browne

Published Oct 1, 1997• 18 citations

Abstract

We study the phase diagram and critical behavior of an interacting one-dimensional two-species monomer-monomer catalytic surface reaction model with a reactive phase as well as two equivalent adsorbing phases where one of the species saturates the system. The model depends on two parameters: the relative adsorption rates of the two species and a repulsive interaction between like neighbors. A mean-field analysis including correlations up to triplets of sites fails to reproduce the phase diagram found by Monte Carlo simulations. The three phases coexist at a bicritical point whose critical behavior is described by the even branching annihilating random-walk universality class. This work confirms the hypothesis that the conservation modulo 2 of the domain walls under the dynamics at the bicritical point is the essential feature in producing critical behavior different from directed percolation. The interfacial fluctuations show the same universal behavior seen at the bicritical point in a three-species model, supporting the conjecture that these fluctuations are an additional universal characteristic of the model.

DOI: 10.1103/physreve.56.3953

Related papers

Critical Behavior of an Interacting Monomer-Dimer Model

Mann Ho Kim, Hyunggyu Park

We study a monomer-dimer model with repulsive interactions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. Monte Carlo simulations show that the critical behavior is different from the conventional directed percolation universality class but seems to be consistent with that of the models with the mass conservation of modulo 2.

Physical Review Letters•Nov 7, 1994•125 citations

Critical behavior of an interacting monomer-dimer model

Hyunggyu Park

We study a monomer-dimer model with repulsive interactions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. Monte Carlo simulations show that the critical behavior is different from the conventional directed percolation universality class but seems to be consistent with that of the models with the mass conservation of modulo 2.

arXiv:cond-mat/9312048•Dec 13, 1993•0 citations

Dynamic scaling behavior of an interacting monomer-dimer model

Heungwon Park, Mann Ho Kim, Hyunggyu Park

We study the dynamic scaling behavior of a monomer-dimer model with repulsive itneractions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. This transition does not belong to the conventional directed percolation universality class. The values of dynamic scaling exponents are estimated by Monte Carlo simulations for two distinct initial configurations, one near an absorbing state and the other with an interface between two different absorbing states. We confirm that the critical behavior is consistent with that of the models with the mass conservation of modulo 2.

arXiv:cond-mat/9508071•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 1, 1995•53 citations

Surface critical behavior in systems with nonequilibrium phase transitions

Martin Howard, Per Fröjdh, K. B. Lauritsen

We study the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d(c), with d(c)=4 (DP) and d(c)=2 (BARW), we use mean field-theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension d</=d(c), we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field-theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1 and 2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta(1) exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta(1) exponents at the ordinary and special transitions. Many of our predictions are supported using Monte Carlo simulations of two different models belonging to the BARW universality class.

arXiv:cond-mat/9907446•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 1, 2000•13 citations

Dynamic phase transitions in diffusion-limited reactions

Uwe Claus Täuber

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.

arXiv:cond-mat/0205327•May 15, 2002•3 citations

Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

Yunqing Ouyang, Youjin Deng, Henk W. J. Blöte

We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number $z$ of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite $z$ are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for $q=3$ and $q=4$ Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.

arXiv:1808.05812•Physical review. E•Dec 3, 2018•22 citations

We study the phase diagram and critical behavior of an interacting one-dimensional two-species monomer-monomer catalytic surface reaction model with a reactive phase as well as two equivalent adsorbing phases where one of the species saturates the system. The model depends on two parameters: the relative adsorption rates of the two species and a repulsive interaction between like neighbors. A mean-field analysis including correlations up to triplets of sites fails to reproduce the phase diagram found by Monte Carlo simulations. The three phases coexist at a bicritical point whose critical behavior is described by the even branching annihilating random-walk universality class. This work confirms the hypothesis that the conservation modulo 2 of the domain walls under the dynamics at the bicritical point is the essential feature in producing critical behavior different from directed percolation. The interfacial fluctuations show the same universal behavior seen at the bicritical point in a three-species model, supporting the conjecture that these fluctuations are an additional universal characteristic of the model.

We study the dynamic scaling behavior of a monomer-dimer model with repulsive itneractions between the same species in one dimension. With infinitely strong interactions the model exhibits a continuous transition from a reactive phase to an inactive phase with two equivalent absorbing states. This transition does not belong to the conventional directed percolation universality class. The values of dynamic scaling exponents are estimated by Monte Carlo simulations for two distinct initial configurations, one near an absorbing state and the other with an interface between two different absorbing states. We confirm that the critical behavior is consistent with that of the models with the mass conservation of modulo 2.

We study the surface critical behavior of branching-annihilating random walks with an even number of offspring (BARW) and directed percolation (DP) using a variety of theoretical techniques. Above the upper critical dimensions d(c), with d(c)=4 (DP) and d(c)=2 (BARW), we use mean field-theory to analyze the surface phase diagrams using the standard classification into ordinary, special, surface, and extraordinary transitions. For the case of BARW, at or below the upper critical dimension d</=d(c), we use field theoretic methods to study the effects of fluctuations. As in the bulk, the field-theory suffers from technical difficulties associated with the presence of a second critical dimension. However, we are still able to analyze the phase diagrams for BARW in d=1 and 2, which turn out to be very different from their mean field analog. Furthermore, for the case of BARW only (and not for DP), we find two independent surface beta(1) exponents in d=1, arising from two distinct definitions of the order parameter. Using an exact duality transformation on a lattice BARW model in d=1, we uncover a relationship between these two surface beta(1) exponents at the ordinary and special transitions. Many of our predictions are supported using Monte Carlo simulations of two different models belonging to the BARW 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.

We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number

z

of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite

z

are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for

q=3

and

q=4

Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.