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Title: Discontinuous non-equilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic two-phase coexistence and metastability

Abstract

Threshold versions of Schloegl’s model on a lattice, which involve autocatalytic creation and spontaneous annihilation of particles, can provide a simple prototype for discontinuous non-equilibrium phase transitions. These models are equivalent to so-called threshold contact processes. A discontinuous transition between populated and vacuum states can occur selecting a threshold of N ≥ 2 for the minimum number, N, of neighboring particles enabling autocatalytic creation at an empty site. Fundamental open questions remain given the lack of a thermodynamic framework for analysis. For a square lattice with N = 2, we show that phase coexistence occurs not at a unique value but for a finite range of particle annihilation rate (the natural control parameter). This generic two-phase coexistence also persists when perturbing the model to allow spontaneous particle creation. Such behavior contrasts both the Gibbs phase rule for thermodynamic systems and also previous analysis for this model. We find metastability near the transition corresponding to a non-zero effective line tension, also contrasting previously suggested critical behavior. Mean-field type analysis, extended to treat spatially heterogeneous states, further elucidates model behavior.

Authors:
 [1];  [1];  [1]
  1. Ames Laboratory–USDOE, Iowa State University, Ames, Iowa 50011 (United States)
Publication Date:
OSTI Identifier:
22415697
Resource Type:
Journal Article
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 142; Journal Issue: 16; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; ANNIHILATION; EQUILIBRIUM; MEAN-FIELD THEORY; PARTICLES; PHASE RULE; PHASE STABILITY; PHASE STUDIES; PHASE TRANSFORMATIONS; TETRAGONAL LATTICES; THERMODYNAMICS; VACUUM STATES

Citation Formats

Wang, Chi-Jen, Department of Mathematics, Iowa State University, Ames, Iowa 50011, Liu, Da-Jiang, Evans, James W., Department of Mathematics, Iowa State University, Ames, Iowa 50011, and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011. Discontinuous non-equilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic two-phase coexistence and metastability. United States: N. p., 2015. Web. doi:10.1063/1.4918908.
Wang, Chi-Jen, Department of Mathematics, Iowa State University, Ames, Iowa 50011, Liu, Da-Jiang, Evans, James W., Department of Mathematics, Iowa State University, Ames, Iowa 50011, & Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011. Discontinuous non-equilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic two-phase coexistence and metastability. United States. https://doi.org/10.1063/1.4918908
Wang, Chi-Jen, Department of Mathematics, Iowa State University, Ames, Iowa 50011, Liu, Da-Jiang, Evans, James W., Department of Mathematics, Iowa State University, Ames, Iowa 50011, and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011. 2015. "Discontinuous non-equilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic two-phase coexistence and metastability". United States. https://doi.org/10.1063/1.4918908.
@article{osti_22415697,
title = {Discontinuous non-equilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic two-phase coexistence and metastability},
author = {Wang, Chi-Jen and Department of Mathematics, Iowa State University, Ames, Iowa 50011 and Liu, Da-Jiang and Evans, James W. and Department of Mathematics, Iowa State University, Ames, Iowa 50011 and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011},
abstractNote = {Threshold versions of Schloegl’s model on a lattice, which involve autocatalytic creation and spontaneous annihilation of particles, can provide a simple prototype for discontinuous non-equilibrium phase transitions. These models are equivalent to so-called threshold contact processes. A discontinuous transition between populated and vacuum states can occur selecting a threshold of N ≥ 2 for the minimum number, N, of neighboring particles enabling autocatalytic creation at an empty site. Fundamental open questions remain given the lack of a thermodynamic framework for analysis. For a square lattice with N = 2, we show that phase coexistence occurs not at a unique value but for a finite range of particle annihilation rate (the natural control parameter). This generic two-phase coexistence also persists when perturbing the model to allow spontaneous particle creation. Such behavior contrasts both the Gibbs phase rule for thermodynamic systems and also previous analysis for this model. We find metastability near the transition corresponding to a non-zero effective line tension, also contrasting previously suggested critical behavior. Mean-field type analysis, extended to treat spatially heterogeneous states, further elucidates model behavior.},
doi = {10.1063/1.4918908},
url = {https://www.osti.gov/biblio/22415697}, journal = {Journal of Chemical Physics},
issn = {0021-9606},
number = 16,
volume = 142,
place = {United States},
year = {Tue Apr 28 00:00:00 EDT 2015},
month = {Tue Apr 28 00:00:00 EDT 2015}
}

Works referenced in this record:

“First-Principles” kinetic monte carlo simulations revisited: CO oxidation over RuO2(110)
journal, January 2012


On phase transitions in Schl�gl's second model
journal, December 1982


Chemical reaction models for non-equilibrium phase transitions
journal, April 1972


Wulff Droplets and the Metastable Relaxation of Kinetic Ising Models
journal, June 1998


Threshold θ ≥ 2 contact processes on homogeneous trees
journal, July 2007


Schloegl’s Second Model for Autocatalysis on a Cubic Lattice: Mean-Field-Type Discrete Reaction-Diffusion Equation Analysis
journal, September 2011


Kinetic phase transitions in a model for surface catalysis
journal, October 1989


Two versions of the threshold contact model in two dimensions
journal, September 2012


Generic two-phase coexistence, relaxation kinetics, and interface propagation in the quadratic contact process: Analytic studies
journal, January 2008


A first order phase transition in the threshold <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>θ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math> contact process on random <mml:math altimg="si2.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>r</mml:mi></mml:math>-regular graphs and <mml:math altimg="si3.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>r</mml:mi></mml:math>-trees
journal, February 2013


Metastable states: smooth continuations through the critical point
journal, February 1999


Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model
journal, January 2004


Kinetics of Phase Change. I General Theory
journal, December 1939


Kinetics of Phase Change. II Transformation‐Time Relations for Random Distribution of Nuclei
journal, February 1940


Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III
journal, February 1941


Schloegl’s second model for autocatalysis with particle diffusion: Lattice-gas realization exhibiting generic two-phase coexistence
journal, February 2009


Transitions between strongly correlated and random steady-states for catalytic CO-oxidation on surfaces at high-pressure
journal, April 2015


Stochastic Spatial Models of Host-Pathogen and Host-Mutualist Interactions II
journal, August 2010


Nonlinearity in complexity science
journal, November 2008


Critical discontinuous phase transition in the threshold contact process
journal, March 2011


Kinetic phase transitions in a surface-reaction model: Mean-field theory
journal, November 1986


Generic nonergodic behavior in locally interacting continuous systems
journal, September 1990


Metastable lifetimes in a kinetic Ising model: Dependence on field and system size
journal, June 1994


Trigger waves in a model for catalysis
journal, December 1995


Propagating waves in one-dimensional discrete networks of coupled units
journal, March 2004


Decay of metastable phases in a model for the catalytic oxidation of CO
journal, March 2005


Contact process with long-range interactions: A study in the ensemble of constant particle number
journal, October 2007


Robustness of first-order phase transitions in one-dimensional long-range contact processes
journal, April 2013


Role of Irreversibility in Stabilizing Complex and Nonergodic Behavior in Locally Interacting Discrete Systems
journal, August 1985


Kinetic Phase Transitions in an Irreversible Surface-Reaction Model
journal, June 1986


Characterizing kinetics near a first-order catalytic-poisoning transition
journal, February 1991


Nonequilibrium Model for the Contact Process in an Ensemble of Constant Particle Number
journal, June 2001


Critical Coarsening without Surface Tension: The Universality Class of the Voter Model
journal, July 2001


Quadratic Contact Process: Phase Separation with Interface-Orientation-Dependent Equistability
journal, February 2007


Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells
journal, June 1987


Stochastic Spatial Models
journal, January 1999


Generic two-phase coexistence in nonequilibrium systems
journal, January 2005


Dynamics of Lattice Differential Equations
journal, September 1996


The survival of large dimensional threshold contact processes
journal, July 2009


“First-Principles” kinetic monte carlo simulations revisited: CO oxidation over RuO2(110)
text, January 2012


Nonequilibrium Model for the Contact Process in an Ensemble of Constant Particle Number
text, January 2001


Kinetic phase transitions in a model for surface catalysis
journal, October 1989


The survival of large dimensional threshold contact processes
text, January 2009


Generic two-phase coexistence in nonequilibrium systems
text, January 2004


Works referencing / citing this record: