Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems
Abstract
Bistable nonequilibrium systems are realized in catalytic reactiondiffusion processes, biological transport and regulation, spatial epidemics, etc. Behavior in spatially continuous formulations, described at the meanfield level by reactiondiffusion type equations (RDEs), often mimics that of classic equilibrium van der Waals type systems. When accounting for noise, similarities include a discontinuous phase transition at some value, ${p}_{\mathrm{eq}}$, of a control parameter, $p$, with metastability and hysteresis around ${p}_{\mathrm{eq}}$. For each $p$, there is a unique critical droplet of the more stable phase embedded in the less stable or metastable phase which is stationary (neither shrinking nor growing), and with size diverging as $p\to {p}_{\mathrm{eq}}$. Spatially discrete analogs of these meanfield formulations, described by lattice differential equations (LDEs), are more appropriate for some applications, but have received less attention. It is recognized that LDEs can exhibit richer behavior than RDEs, specifically propagation failure for planar interphases separating distinct phases. Herein, we show that this feature, together with an orientation dependence of planar interface propagation also deriving from spatial discreteness, results in the occurrence of entire families of stationary droplets. The extent of these families increases approaching the transition and can be infinite if propagation failure is realized. In addition, there can exist a regime of generic twophase coexistence where arbitrarily large droplets of either phase always shrink. Such rich behavior is qualitatively distinct from that for classic nucleation in equilibrium and spatially continuous nonequilibrium systems.
 Authors:

 National Chung Cheng Univ., Chiayi (Taiwan)
 Ames Lab., Ames, IA (United States)
 Ames Lab., Ames, IA (United States); Iowa State Univ., Ames, IA (United States)
 Publication Date:
 Research Org.:
 Ames Lab., Ames, IA (United States)
 Sponsoring Org.:
 Ministry of Science and Technology (MOST) of Taiwan; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22). Chemical Sciences, Geosciences & Biosciences Division
 OSTI Identifier:
 1602867
 Alternate Identifier(s):
 OSTI ID: 1602361
 Report Number(s):
 ISJ10048
Journal ID: ISSN 24700045; PLEEE8
 Grant/Contract Number:
 1052115M194011MY2; AC0207CH11358
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review E
 Additional Journal Information:
 Journal Volume: 101; Journal Issue: 2; Journal ID: ISSN 24700045
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Wang, ChiJen, Liu, DaJiang, and Evans, James W. Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems. United States: N. p., 2020.
Web. doi:10.1103/PhysRevE.101.022803.
Wang, ChiJen, Liu, DaJiang, & Evans, James W. Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems. United States. doi:https://doi.org/10.1103/PhysRevE.101.022803
Wang, ChiJen, Liu, DaJiang, and Evans, James W. Fri .
"Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems". United States. doi:https://doi.org/10.1103/PhysRevE.101.022803. https://www.osti.gov/servlets/purl/1602867.
@article{osti_1602867,
title = {Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems},
author = {Wang, ChiJen and Liu, DaJiang and Evans, James W.},
abstractNote = {Bistable nonequilibrium systems are realized in catalytic reactiondiffusion processes, biological transport and regulation, spatial epidemics, etc. Behavior in spatially continuous formulations, described at the meanfield level by reactiondiffusion type equations (RDEs), often mimics that of classic equilibrium van der Waals type systems. When accounting for noise, similarities include a discontinuous phase transition at some value, peq, of a control parameter, p, with metastability and hysteresis around peq. For each p, there is a unique critical droplet of the more stable phase embedded in the less stable or metastable phase which is stationary (neither shrinking nor growing), and with size diverging as p→peq. Spatially discrete analogs of these meanfield formulations, described by lattice differential equations (LDEs), are more appropriate for some applications, but have received less attention. It is recognized that LDEs can exhibit richer behavior than RDEs, specifically propagation failure for planar interphases separating distinct phases. Herein, we show that this feature, together with an orientation dependence of planar interface propagation also deriving from spatial discreteness, results in the occurrence of entire families of stationary droplets. The extent of these families increases approaching the transition and can be infinite if propagation failure is realized. In addition, there can exist a regime of generic twophase coexistence where arbitrarily large droplets of either phase always shrink. Such rich behavior is qualitatively distinct from that for classic nucleation in equilibrium and spatially continuous nonequilibrium systems.},
doi = {10.1103/PhysRevE.101.022803},
journal = {Physical Review E},
number = 2,
volume = 101,
place = {United States},
year = {2020},
month = {2}
}
Works referenced in this record:
From atomistic latticegas models for surface reactions to hydrodynamic reactiondiffusion equations
journal, March 2002
 Evans, J. W.; Liu, DaJiang; Tammaro, M.
 Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 12, Issue 1
Universality of Crystallographic Pinning
journal, February 2010
 Hoffman, Aaron; MalletParet, John
 Journal of Dynamics and Differential Equations, Vol. 22, Issue 2
Transitions between strongly correlated and random steadystates for catalytic COoxidation on surfaces at highpressure
journal, April 2015
 Liu, DaJiang; Evans, James W.
 The Journal of Chemical Physics, Vol. 142, Issue 13
Kinetic phase transitions in catalytic reaction models
journal, November 1991
 Evans, J. W.
 Langmuir, Vol. 7, Issue 11
FirstPrinciples Statistical Mechanics Study of the Stability of a Subnanometer Thin Surface Oxide in Reactive Environments: CO Oxidation at Pd(100)
journal, January 2007
 Rogal, Jutta; Reuter, Karsten; Scheffler, Matthias
 Physical Review Letters, Vol. 98, Issue 4
SelfOrganized Stationary Patterns in Networks of Bistable Chemical Reactions
journal, October 2016
 Kouvaris, Nikos E.; Sebek, Michael; Mikhailov, Alexander S.
 Angewandte Chemie International Edition, Vol. 55, Issue 42
Tunable Pinning of Burst Waves in Extended Systems with Discrete Sources
journal, December 1998
 Mitkov, Igor; Kladko, Konstantin; Pearson, John E.
 Physical Review Letters, Vol. 81, Issue 24
CO oxidation on Pd(100) at technologically relevant pressure conditions: Firstprinciples kinetic Monte Carlo study
journal, April 2008
 Rogal, Jutta; Reuter, Karsten; Scheffler, Matthias
 Physical Review B, Vol. 77, Issue 15
Traveling Wave Solutions for Systems of ODEs on a TwoDimensional Spatial Lattice
journal, January 1998
 Van Vleck, Erik S.; MalletParet, John; Cahn, John W.
 SIAM Journal on Applied Mathematics, Vol. 59, Issue 2
Absence of Diffusion in Certain Random Lattices
journal, March 1958
 Anderson, P. W.
 Physical Review, Vol. 109, Issue 5
Bistability in Pulse Propagation in Networks of Excitatory and Inhibitory Populations
journal, April 2001
 Golomb, David; Ermentrout, G. Bard
 Physical Review Letters, Vol. 86, Issue 18
Allee effects in biological invasions
journal, August 2005
 Taylor, Caz M.; Hastings, Alan
 Ecology Letters, Vol. 8, Issue 8
Quadratic Contact Process: Phase Separation with InterfaceOrientationDependent Equistability
journal, February 2007
 Liu, DaJiang; Guo, Xiaofang; Evans, J. W.
 Physical Review Letters, Vol. 98, Issue 5
Spatially Discrete FitzHughNagumo Equations
journal, January 2005
 Elmer, Christopher E.; Van Vleck, Erik S.
 SIAM Journal on Applied Mathematics, Vol. 65, Issue 4
Kinetic Phase Transitions in an Irreversible SurfaceReaction Model
journal, June 1986
 Ziff, Robert M.; Gulari, Erdagon; Barshad, Yoav
 Physical Review Letters, Vol. 56, Issue 24
Dynamics of Lattice Differential Equations
journal, September 1996
 Chow, ShuiNee; MalletParet, John; Van Vleck, Erik S.
 International Journal of Bifurcation and Chaos, Vol. 06, Issue 09
Theory of crystal growth and interface motion in crystalline materials
journal, August 1960
 Cahn, John W.
 Acta Metallurgica, Vol. 8, Issue 8
Chemical reaction models for nonequilibrium phase transitions
journal, April 1972
 Schl�gl, F.
 Zeitschrift f�r Physik, Vol. 253, Issue 2
Orientation‐Dependent Dissolution of Germanium
journal, November 1960
 Frank, F. C.; Ives, M. B.
 Journal of Applied Physics, Vol. 31, Issue 11
Schloegl's second model for autocatalysis on hypercubic lattices: Dimension dependence of generic twophase coexistence
journal, April 2012
 Wang, ChiJen; Liu, DaJiang; Evans, J. W.
 Physical Review E, Vol. 85, Issue 4
Chemical diffusion in the lattice gas of noninteracting particles
journal, January 1981
 Kutner, R.
 Physics Letters A, Vol. 81, Issue 4
Propagation failure of traveling waves in a discrete bistable medium
journal, May 1998
 Fáth, Gábor
 Physica D: Nonlinear Phenomena, Vol. 116, Issue 12
Kinetics of Phase Change. I General Theory
journal, December 1939
 Avrami, Melvin
 The Journal of Chemical Physics, Vol. 7, Issue 12, p. 11031112
Discontinuous nonequilibrium phase transition in a threshold Schloegl model for autocatalysis: Generic twophase coexistence and metastability
journal, April 2015
 Wang, ChiJen; Liu, DaJiang; Evans, James W.
 The Journal of Chemical Physics, Vol. 142, Issue 16
Discontinuous Phase Transitions in Nonlocal Schloegl Models for Autocatalysis: Loss and Reemergence of a Nonequilibrium Gibbs Phase Rule
journal, September 2018
 Liu, DaJiang; Wang, ChiJen; Evans, James W.
 Physical Review Letters, Vol. 121, Issue 12
Kinetic Monte Carlo Simulation of Statistical Mechanical Models and CoarseGrained Mesoscale Descriptions of Catalytic Reaction–Diffusion Processes: 1D Nanoporous and 2D Surface Systems
journal, April 2015
 Liu, DaJiang; Garcia, Andres; Wang, Jing
 Chemical Reviews, Vol. 115, Issue 12
Unidirectional Transition Waves in Bistable Lattices
journal, June 2016
 Nadkarni, Neel; Arrieta, Andres F.; Chong, Christopher
 Physical Review Letters, Vol. 116, Issue 24
Role of Irreversibility in Stabilizing Complex and Nonergodic Behavior in Locally Interacting Discrete Systems
journal, August 1985
 Bennett, Charles H.; Grinstein, G.
 Physical Review Letters, Vol. 55, Issue 7
Schloegl’s second model for autocatalysis with particle diffusion: Latticegas realization exhibiting generic twophase coexistence
journal, February 2009
 Guo, Xiaofang; Liu, DaJiang; Evans, J. W.
 The Journal of Chemical Physics, Vol. 130, Issue 7
Generic twophase coexistence, relaxation kinetics, and interface propagation in the quadratic contact process: Analytic studies
journal, January 2008
 Guo, Xiaofang; Evans, J. W.; Liu, DaJiang
 Physica A: Statistical Mechanics and its Applications, Vol. 387, Issue 1
Stability of Localized Wave Fronts in Bistable Systems
journal, January 2013
 Rulands, Steffen; Klünder, Ben; Frey, Erwin
 Physical Review Letters, Vol. 110, Issue 3
Traveling and Pinned Fronts in Bistable ReactionDiffusion Systems on Networks
journal, September 2012
 Kouvaris, Nikos E.; Kori, Hiroshi; Mikhailov, Alexander S.
 PLoS ONE, Vol. 7, Issue 9
Generic TwoPhase Coexistence and Nonequilibrium Criticality in a Lattice Version of Schlögl’s Second Model for Autocatalysis
journal, March 2009
 Liu, DaJiang
 Journal of Statistical Physics, Vol. 135, Issue 1
Tricriticality in generalized Schloegl models for autocatalysis: Latticegas realization with particle diffusion
journal, February 2012
 Guo, Xiaofang; Unruh, Daniel K.; Liu, DaJiang
 Physica A: Statistical Mechanics and its Applications, Vol. 391, Issue 3