We analyze a discontinuous nonequilibrium phase transition between an active (or reactive) state and a poisoned (or extinguished) state occurring in a stochastic lattice-gas realization of Schloegl’s second model for autocatalysis. This realization, also known as the quadratic contact process, involves spontaneous annihilation, autocatalytic creation, and diffusion of particles on a square lattice, where creation at empty sites requires a suitable nearby pair of particles. The poisoned state exists for all annihilation rates and is an absorbing particle-free “vacuum” state. The populated active steady state exists only for below a critical value, . If denotes the critical value below which a finite population can survive, then we show that . This strict inequality contrasts a postulate of Durrett, and is a direct consequence of the occurrence of coexisting stable active and poisoned states for a finite range (which shrinks with increasing diffusivity). This so-called generic two-phase coexistence markedly contrasts behavior in thermodynamic systems. However, one still finds metastability and nucleation phenomena similar to those in discontinuous equilibrium transitions.
REFERENCES
1.
G.
Nicolis
and I.
Prigogine
, Self-Organization in Non-Equilibrium Systems
(Wiley
, New York
, 1977
).2.
P.
Gray
and S. K.
Scott
, Chemical Oscillations and Instabilities
(Clarendon
, Oxford
, 1994
).3.
A. S.
Mikhailov
, Foundations of Synergetics I
(Springer
, Berlin
, 1990
).4.
Chemical Waves and Patterns
, edited by R.
Kapral
and K.
Showalter
(Kluwer
, Amsterdam
, 1995
).5.
N. G.
Van Kampen
, Stochastic Processes in Physics and Chemistry
(North Holland
, Amsterdam
, 1981
).6.
A. S.
Mikhailov
and A. Yu.
Loskutov
, Foundations of Synergetics II
(Springer
, Berlin
, 1996
).7.
J.
Garcia-Ojalvo
and J. M.
Sancho
, Noise in Spatially Extended Systems
(Springer
, Berlin
, 1999
).8.
M.
Hildebrand
, A. S.
Mikhailov
, and G.
Ertl
, Phys. Rev. Lett.
81
, 2602
(1998
).9.
D. T.
Gillespie
and L.
Petzold
, in System Modeling in Cellular Biology
, edited by Z.
Szallasi
, J.
Stelling
, and V.
Periwal
(MIT Press
, Cambridge
, 2006
), p. 331
.10.
R.
Imbihl
, Prog. Surf. Sci.
44
, 185
(1993
).11.
R.
Imbihl
and G.
Ertl
, Chem. Rev. (Washington, D.C.)
95
, 697
(1995
).12.
Y.
Suchorski
, J.
Beben
, E. W.
James
, J. W.
Evans
, and R.
Imbihl
, Phys. Rev. Lett.
82
, 1907
(1999
).13.
D. -J.
Liu
and J. W.
Evans
, J. Phys.: Condens. Matter
19
, 065129
(2007
).14.
J.
Starke
, C.
Reichert
, M.
Eiswirth
, H. H.
Rotermund
, and G.
Ertl
, Europhys. Lett.
73
, 820
(2006
).15.
D.
Dab
, A.
Lawniczak
, J. P.
Boon
, and R.
Kapral
, Phys. Rev. Lett.
64
, 2462
(1990
).16.
J. P.
Boon
, D.
Dab
, R.
Kapral
, and A.
Lawniczak
, Phys. Rep.
273
, 55
(1996
).17.
J.
Marro
and R.
Dickman
, Nonequilibrium Phase Transitions in Lattice Models
(Cambridge University Press
, Cambridge
, 1999
).18.
H.
Hinrichsen
, Adv. Phys.
49
, 815
(2000
).19.
G.
Odor
, Rev. Mod. Phys.
76
, 663
(2004
).20.
R. M.
Ziff
, E.
Gulari
, and Y.
Barshad
, Phys. Rev. Lett.
56
, 2553
(1986
).21.
J. W.
Evans
and T. R.
Ray
, Phys. Rev. E
50
, 4302
(1994
).22.
R. H.
Goodman
, D. S.
Graff
, L. M.
Sander
, P.
Leroux-Hugon
, and E.
Clément
, Phys. Rev. E
52
, 5904
(1995
).23.
J. W.
Evans
and M. S.
Miesch
, Phys. Rev. Lett.
66
, 833
(1991
).24.
E.
Loscar
and E. V.
Albano
, Rep. Prog. Phys.
66
, 1343
(2003
).25.
E.
Machado
, G. M.
Buendia
, and P. A.
Rikvold
, Phys. Rev. E
71
, 031603
(2005
).26.
D. -J.
Liu
and J. W.
Evans
, Surf. Sci.
, DOI:10.1016/j.susc.2008.10.058 (2009).27.
F.
Schlögl
, Z. Phys.
253
, 147
(1972
).28.
P.
Grassberger
, Z. Phys. B: Condens. Matter
47
, 365
(1982
).29.
R.
Durrett
, SIAM Rev.
41
, 677
(1999
).30.
D. -J.
Liu
, X.
Guo
, and J. W.
Evans
, Phys. Rev. Lett.
98
, 050601
(2007
).31.
X.
Guo
, D. -J.
Liu
, and J. W.
Evans
, Phys. Rev. E
75
, 061129
(2007
).32.
33.
A. L.
Toom
, in Multicomponent Random Systems
, edited by R. L.
Dobrushin
and Y. G.
Sinai
(Dekker
, New York
, 1980
).34.
C. H.
Bennett
and G.
Grinstein
, Phys. Rev. Lett.
55
, 657
(1985
).35.
36.
M. A.
Munoz
, F.
de los Santos
, and M. M. T.
da Gama
, Eur. Phys. J. B
43
, 73
(2005
).37.
38.
39.
X.
Guo
, D. -J.
Liu
, and J. W.
Evans
, “Schloegl's second model for autocatalysis on a square lattice with particle diffusion: analytic treatment
.”40.
Interchange the role of particles and empty sites in the QCP (Refs. 30 and 31). Then, one has spontaneous creation of particles at empty sites, and their autocatalytic removal given suitable empty pairs of sites. The former mimics monomer adsorption in the ZGB model, and the latter mimics reactive monomer removal (which requires empty pairs sites to allow dimer adsorption).
41.
R. M.
Ziff
and B. J.
Brosilow
, Phys. Rev. A
46
, 4630
(1992
).42.
D. -J.
Liu
, J. Stat. Phys.
, “Generic two-phase coexistence and non-equilibrium criticality in a lattice version of Schloegl's second model for autocatalysis.”43.
R. H.
Schonmann
and S. B.
Shlosman
, Commun. Math. Phys.
194
, 389
(1998
).44.
S.
Shlosman
, Physica A
263
, 180
(1999
).45.
S.
Friedli
and C. -E.
Pfister
, Phys. Rev. Lett.
92
, 015702
(2004
).46.
P. A.
Rikvold
, H.
Tomita
, S.
Miyashita
, and S. W.
Sides
, Phys. Rev. E
49
, 5080
(1994
). This study described the spinodal associated with an infinite Ising system as the “mean-field spinodal point” (MFSP) although it cannot be assessed with a simple mean-field theory.47.
M.
Avrami
, J. Chem. Phys.
7
, 1103
(1939
);M.
Avrami
, J. Chem. Phys.
8
, 212
(1940
);M.
Avrami
, J. Chem. Phys.
9
, 177
(1941
).48.
J.
Krug
and H.
Spohn
, in Solids Far from Equilibrium: Growth, Morphology and Defects
, edited by C.
Godreche
(Cambridge University Press
, Cambridge
, 1991
).49.
A. -L.
Barabasi
and H. E.
Stanley
, Fractal Concepts in Surface Growth
(Cambridge University Press
, Cambridge
, 1995
).50.
For a planar interface with small slope measured relative to , the normal velocity is denoted . By a Pythagorean construction, the velocity in the -direction is , where , and (Ref. 31). Thus, one has .
51.
H. -C.
Jeong
and E. D.
Williams
, Surf. Sci. Rep.
34
, 171
(1999
).52.
Writing , it follows that . These identities are equivalent to Eqs. (15) and (22) noting that and .
53.
An increasing number of vacuum clusters which nucleate just ahead of the interface can be incorporated into the advancing interface as time progresses corrupting estimation of .
54.
J. D.
Gunton
and M.
Droz
, Introduction to the Theory of Metastable and Unstable States, Springer Lecture Notes in Physics
(Springer
, Berlin
, 1983
), Vol. 183
.55.
A.
Matzavinos
, X.
Guo
, D. -J.
Liu
, and J. W.
Evans
. “Analysis of coarse-grained stochastic reaction-diffusion equations for Schloegl's second model on a square lattice. Here, is the nonconserved particle annihilation-creation noise, and is the conserved particle diffusion noise. and are independent white noises.”
56.
M.
Hildebrand
and A. S.
Mikhailov
, J. Phys. Chem.
100
, 19089
(1996
).57.
D. -J.
Liu
and J. W.
Evans
, “Generic two-phase coexistence in lattice-gas reaction-diffusion models.
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