We study a three species monomer-monomer catalytic surface reaction model
with a reactive steady state bordered by three equivalent unreactive phases
where the surface is saturated with one species. The transition from the
reactive to a saturated phase shows directed percolation critical behavior.
Each pair of these reactive-saturated phase boundaries join at a bicritical
point where the universal behavior is in the even branching annihilating random
walk class. We find the crossover exponent from bicritical to critical behavior
and a new exponent associated with the bicritical interface dynamics.Comment: 4 pages RevTex. 4 eps figures included with psfig.sty. Uses
  multicol.sty. Accepted for publication in PR

A. A. Lushnikov

Dana A. Browne

G. Grinstein

H. K. Janssen

H. Park

H. Takayasu

I. Jensen

J. F. F. Mendes

J. G. Amar

J. L. Cardy

J. L. Spouge

J. Marro

Kevin E. Bassler

M. E. Fisher

M. H. Kim

N. Menyhárd

P. Grassberger

R. Dickman

R. M. Ziff

T. Aukrust

T. E. Harris

English

arXiv

Crossref

Physical Review Letters

Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer Model

We study a three species monomer-monomer catalytic surface reaction model with a reactive steady state bordered by three equivalent unreactive phases where the surface is saturated with one species. The transition from the reactive to a saturated phase shows directed percolation critical behavior. Each pair of these reactive-saturated phase boundaries joins at a bicritical point where the universal behavior is in the even branching annihilating random walk class. We find the crossover exponent from bicritical to critical behavior and a new exponent associated with the bicritical interface dynamics. © 1996 The American Physical Society

Bassler, Kevin E.

Browne, Dana A.

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 11-4-1996 Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer Model Kevin E. Bassler Louisiana State University Dana A. Browne Louisiana State University Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Bassler, K., & Browne, D. (1996). Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer Model. Physical Review Letters, 77 (19), 4094-4097. https://doi.org/10.1103/PhysRevLett.77.4094 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:cond-mat/9609191v1  19 Sep 1996Nonequilibrium critical dynamics of a three species monomer-monomer modelKevin E. Bassler and Dana A. BrowneDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803(June 11, 1996; revised August 28, 1996)We study a three species monomer-monomer catalytic surface reaction model with a reactivesteady state bordered by three equivalent unreactive phases where the surface is saturated with onespecies. The transition from the reactive to a saturated phase shows directed percolation criticalbehavior. Each pair of these reactive-saturated phase boundaries join at a bicritical point where theuniversal behavior is in the even branching annihilating random walk class. We find the crossoverexponent from bicritical to critical behavior and a new exponent associated with the bicriticalinterface dynamics.05.70.Ln, 82.20.Mj, 82.65.Jv, 64.50.HtNonequilibrium models with many degrees of freedomwhose dynamics violate detailed balance arise in studiesof biological populations, heterogeneous catalysis, fluidturbulence, and elsewhere. The macroscopic behavior ofthese models can be much richer than that of systemsin thermal equilibrium, showing organized macroscopicspatial and temporal structures like pulses or waves, andeven spatiotemporal chaos. Even the steady state be-havior can be far more complicated, involving for ex-ample generic scale invariance. Like their equilibriumcousins, nonequilibrium systems at continuous transi-tions between steady states show universal behavior thatis insensitive to microscopic details and depends only onproperties such as symmetries and conservation laws.One of the most common continuous phase transitionsin nonequilibrium models is a transition to an absorbing,noiseless steady state [1], the term absorbing indicatingthe state cannot be left once it is reached. Examplesof this include directed percolation (DP) [2,3], the con-tact process [4], the dimer poisoning transition in theZGB model [5] for the catalytic oxidation of CO, auto-catalytic reaction models [6], and branching annihilatingrandom walks with odd numbers of offspring [7,8]. Bothrenormalization group calculations [2,9] and Monte Carlosimulations [3–8] show that these models form a singleuniversality class for a purely nonequilibrium model withno internal symmetry in the order parameter.Recently, a number of models with continuous adsorb-ing transitions in a universality class distinct from di-rected percolation have been studied. These models in-clude the probabilistic cellular automata models studiedof Grassberger et al. [10], certain kinetic Ising models[11], the interacting monomer-dimer model [12,13], andbranching annihilating random walks with an even num-ber of offspring (BAWe) [7,14]. All of these models ex-cept for the BAWe have two equivalent absorbing statesindicating the importance of symmetry of the adsorbingstate to the universality class. However, the universalbehavior of this new class is apparently controlled by adynamical conservation law. If [10] the important dy-namical variables in this class are defects represented bythe walkers in the BAWe model and the walls between dif-ferent saturated domains in the other models, the modelshave “defect parity” conservation law where the numberof defects is conserved modulo 2. Recent field theoreticwork confirms this viewpoint [15].We study here a simple catalytic surface reactionmodel with three different equivalent monomer species.This model could represent either a system with threedifferent chemical species or an auto-catalytic reactionsystem in which one chemical species can adsorb on threedifferent types of surface sites. This model has adsorb-ing transitions to both one and two equivalent noiselessstates, and therefore is a good model to study the roleof symmetry in adsorbing phase transitions. The modelhas two fundamental processes: (a) monomer adsorptiononto sites of a substrate, and (b) the annihilation reactionof two dissimilar monomers adsorbed on nearest-neighborsites of the substrate. Here we consider the model only inthe adsorption controlled limit where process (b) occursinstantaneously. We present here results only for the onedimensional version of the model.Calling the monomer species A, B and C, the param-eters in the model are then the relative adsorption ratesof the different monomer species pA, pB, and pC , suchthat pA + pB + pC = 1. Using static Monte Carlo sim-ulations to find the steady state, and dynamical MonteCarlo studies described below, we find the ternary phasediagram for the model is shown in Fig. 1, where the hor-izontal axis corresponds to the relative adsorption rateof A and B monomers pAB = pA/(pA + pB). There arenoiseless phases where one monomer species saturates thechain occupying the corners of the phase diagram and areactive steady state in the center. There are continuousphase transitions from the reactive phase to the satu-rated phases. The monomer densities undergo discon-tinuous, first-order, transitions from one saturated stateto another. The points where the reactive phase and two1saturated phases meet are bicritical points [16] where twolines of continuous transitions meet a line of first ordertransitions. We have also constructed a mean field the-ory of the model, to be presented elsewhere [17], followingthe methods of Dickman [18]. One unusual feature of themean field phase diagram is that the bicritical points lieon the edge of the phase diagram if the correlations arecorrect up to single sites or even nearest neighbor pairs.Only when the correlations up to triplets of adjacent sitesare included does the bicritical point appear inside thephase diagram, indicating the importance of reproduc-ing the correlations induced by large domains of a singlesaturated phase.0.0 0.2 0.4 0.6 0.8 1.0pAB0.00.20.40.60.81.0pCB ACFIG. 1. Phase diagram showing three saturated phases(indicated by the letters), and a reactive phase (the unla-beled center region). Solid lines indicate continuous transi-tions. Dashed lines indicate first-order transitions.We have used dynamical Monte Carlo simulations toinvestigate the universality classes of the continuous tran-sitions and bicritical points, the critical dynamics of in-terfaces between the two symmetric saturated states atthe bicritical points, the crossover behavior near the bi-critical point, and the first-order lines. We use two formsof “epidemic” analysis [8,14,19] following the evolution ofan initial condition chosen close to the saturated state.In the first form (defect dynamics) we use an initial con-dition consisting of a single vacancy in a saturated phase.The second (interface dynamics) starts from two differentsaturated phases separated by a single vacancy.From the simulations we find the probability P (t) thesystem does not fall into the saturated state in t timesteps, the average number of vacancies per run 〈nV (t)〉,and the typical size of the defect or interface per sur-viving run 〈R2(t)〉. At a continuous phase transition ast → ∞ it is expected that they obey power law behaviorP (t) ∼ t−δ, 〈nV (t)〉 ∼ tη, 〈R2(t)〉 ∼ tz. (1)Precise estimates of the location of the critical point andof the exponents can be made by examining the localslopes of the curves on a log-log plot and extrapolatingto infinite times. Then, the effective exponent δ(t) is− δ(t) = {ln [P (t)/P (t/b)] / ln b}, (2)η(t) and z(t) being defined analogously. Plotting the lo-cal slopes versus t−1 allows us to determine both theexponent and the critical point accurately. At the criti-cal point the local slope will extrapolate to a constant ast−1 → 0 with a scaling correction linear in t−1 [20]. Datataken away from the critical point will have local slopesthat curve away from the critical point value as t−1 → 0.0.0 2.5 5.01000/t1.221.261.30z  0.260.310.36η  -0.20-0.16-0.12−δFIG. 2. Effective exponents using Eq. (2) with b = 5 forthe defect dynamics near the critical point at pAB = 0.5 onthe line where the C poisoned phase meets the reactive phase.From top to bottom, the 3 curves in each panel correspondto pC = 0.3955, 0.39575, and 0.3960, with the middle curvecorresponding to the critical point.Figure 2 shows the effective exponents of the three dy-namic quantities near the phase transition to the C satu-rated phase at pAB = 0.5. Using 105 independent runs ofup to 104 time steps at each parameter value, we found acritical C monomer adsorption rate of p̃C = 0.39575(10),and the critical exponents are δ = 0.16(1), η = 0.31(1),and z = 1.255(15). These values are consistent with ourexpectation that this transition should be in the DP uni-versality class, for which the exponents are δ = 0.1596(4),η = 0.3137(10), and z = 1.2660(14) [21]. We found simi-lar exponents for the adsorbing transition at a number ofother points along the lines separating the reactive phaseand the saturated states, indicating the transition be-tween the reactive phase and any single saturated phase2is always in the DP universality class.0.0 2.5 5.010000/t1.101.141.18z  -0.050.000.05η-0.33-0.29-0.25−δFIG. 3. Effective exponents for the defect dynamics nearthe bicritical point where the A and B poisoned phases meetthe reactive phase as defined in (2) with b = 5. From bottomto top, the 3 curves in each panel correspond to pC = 0.121,0.122, and 0.123, with the middle line corresponding to thebicritical point.The same kind of analysis at the bicritical point atpAB = 0.5, using an initial condition of a vacancy inan A-saturated phase, yields a bicritical point at pC =p∗C = 0.122(1), and very different exponents, given thepresence of two-symmetry equivalent saturated phases.From 5 × 105 runs of up to 105 time steps we foundthe local slope data shown in Fig. 3, yielding values ofδ = 0.29(1), η = 0.00(1), and z = 1.150(15). These val-ues indicate that the bicritical behavior falls in the BAWeuniversality class, for which δ = 0.285(2), η = 0.000(1),and z = 1.141(2) [14].For pC < p∗C along the A-B coexistence line, a sim-ilar analysis yields the sub-critical dynamic exponentsδ ≈ 0.5, η ≈ −0.5, and z ≈ 1. These exponents also de-scribe the two species version of the model which can bemapped onto the well known problem of the T = 0 one-dimensional kinetic Ising model for which these valuesare known exactly [22].To further analyze the importance of competition inthe growth of two equivalent saturated phases at the bi-critical point we also studied the dynamics of an interfacebetween those two phases. Starting with a single vacancybetween the two domains, we used two different methodsto analyze the behavior of the interface. Since there mustalways be at least one vacancy between two different sat-urated phases, in the first method we ignore the survivalprobability P (t) and take δ ≡ 0. We then measure thenumber of vacancies in the interface 〈n(t)〉 ∝ tη and aver-age size of the interface 〈R2(t)〉 ∝ tz. From 5× 104 inde-pendent runs at the bicritical point, each lasting 105 timesteps, we found the other exponents to be η = 0.285(10)and z = 1.14(2). This type of interface dynamics hasbeen used to study the properties of critical interfaces inother models in the BAWe class, where similar results forη and z were obtained [13,14].0.0 2.5 5.010000/t1.101.151.20z⁄   0.000 2.500 5.000-0.50-0.43-0.36η ⁄0.000 2.500 5.000-0.78-0.73-0.68−δ⁄FIG. 4. Effective exponents, as in Fig. 3, for the secondtype of interface dynamics near the bicritical point where theA and B poisoned phases meet the reactive phase.In the second type of interface dynamics simulations,which has not been studied before, the simulation isstopped if the interface between the domains has “col-lapsed” back to one vacant site. We introduce a probabil-ity of avoiding a collapse P (t) ∝ t−δ′and correspondingvacancy concentrations 〈n(t)〉 ∝ tη′and 〈R2(t)〉 ∝ tz′.Figure 4 shows results from 107 independent runs eachlasting up to 105 time steps. We find values of δ′ =0.73(2), η′ = −0.43(2) and z′ = 1.15(2).Note the value of the dynamic exponent z or z′, whichmeasures the size of the active region during survivingruns, is the same in both types of interface dynamicssimulations as that measured for the defect dynamics.Furthermore, although the exponents δ and η are differ-ent in the three cases, their sum δ+ η (or δ′ + η′), whichgoverns the time evolution of the number of vacancies injust the surviving runs, seems to be the same. This in-dicates a universal nature of the critical spreading of theactive region for models with two symmetric adsorbingstates which is independent of whether defect or inter-3face dynamics is being considered. A similar result holdsfor some one-dimensional systems with infinitely manyadsorbing states [23].Assuming this conjecture is true, it should be notedthat simulations using the first type of interface dynam-ics, where δ ≡ 0, yield no information beyond that ob-tainable from simulations employing defect dynamics.However, simulations using the second type of interfacedynamics measure an independent dynamic exponent δ′which we expect to be a universal number. It would beinteresting to measure this exponent for other models inthe BAWe class.10-310-210-1100pC - pC*10-510-410-310-210-1p AB - 0.5FIG. 5. Location of the critical line as a function of dis-tance from the bicritical point. The data falls on a line with aslope corresponding to the crossover exponent φ = 2.1± 0.1.Finally, we measured the crossover from bicritical tocritical behavior. Near the bicritical point where the Aand B poisoned phases meet, the boundary of the reactiveregion is expected to behave as (pAB−0.5) ∝ (pC−p∗C)φ,where φ is the crossover exponent [16]. We used the dy-namical simulation method to accurately determine thelocation of the DP phase boundary between the reactivephase and the A saturated phase near the bicritical point.From the log-log plot of pAB − 0.5 versus pC − p∗C shownin Fig. 5 we find φ = 2.1± 0.1.We have introduced a convenient model to study therole of symmetry in the critical dynamics of adsorbingphase transitions. We have shown that the universalityclass of the transition changes from DP to BAWe whenthe symmetry of the adsorbing state is increased from oneto two equivalent noiseless states. Furthermore, we haveshown that having a symmetry in the adsorbing statesintroduces a richness into the dynamics that is not possi-ble if there is a unique adsorbing state. In particular, thecritical dynamics of the interfaces between two differentadsorbing states shows a sensitivity to how the dynam-ics is defined, and the survival probability of fluctuationsin the size of the interface from its smallest value is de-scribed by a new universal exponent δ′. However, thecritical spreading of the reactive region, be it a defectin a single phase or a domain wall between phases, ap-pears to be insensitive to the choice of initial conditions.This appears to result from the fact that large reactiveregions are insensitive to whether the reactive regions arebounded by the same or different saturated phases.This work was supported by the National ScienceFoundation under Grant No. DMR–9408634.[1] J. Marro and R. Dickman, Nonequilibrium phase transi-tions in lattice models (Cambridge Univ. Press, 1996).[2] H. K. Janssen, Z. Phys. B 42, 151 (1981).[3] P. Grassberger, Z. Phys. B 47, 365 (1982).[4] T. E. Harris, Ann. Prob. 2, 969 (1974).[5] R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett.56, 2553 (1986); G. Grinstein, Z.-W. Lai, and D. A.Browne, Phys. Rev. A 40, 4820 (1989); I. Jensen, H.C. Fogedby, and R. Dickman, ibid. 41, 3411 (1990).[6] T. Aukrust, D. A. Browne, and I. Webman, Phys. Rev.A 41, 5294 (1990).[7] H. Takayasu and A. Yu. Tretyakov, Phys. Rev. Lett. 68,3060 (1992).[8] I. Jensen, J. Phys. A 26, 3921 (1993).[9] J. L. Cardy and R. L. Sugar, J. Phys. A 13, L423 (1980).[10] P. Grassberger, F. Krause, and T. von der Twer, J. Phys.A 17, L105 (1984); P. Grassberger, J. Phys. A 22, L1103(1989).[11] N. Menyhárd, J. Phys. A 27, 6139 (1994); N. Menyhárdand G. Ódor, J. Phys. A 27, 6139 (1994).[12] M. H. Kim, and H. Park, Phys. Rev. Lett. 73, 2579(1994); H. Park, and H. Park, Physica A 221, 97 (1995).[13] H. Park, M. H. Kim, and H. Park, Phys. Rev. E 52, 5664(1995).[14] I. Jensen, Phys. Rev. E 50, 3623 (1994).[15] J. Cardy and U. Taüber, preprint.[16] M.E. Fisher and D.R. Nelson, Phys. Rev. Lett. 32, 1350(1974).[17] K. E. Bassler and D. A. Browne, to be published.[18] R. Dickman, Phys. Rev. A 34, 4626 (1986).[19] P. Grassberger, J. Phys. A 22, 3673 (1989); P. Grass-berger and A. de la Torre, Ann. Phys. (New York) 122,373 (1979).[20] Actually, this statement is known to be true only for di-rected percolation [3], but it seems to also be true [14]for the BAWe universality class.[21] I. Jensen and R. Dickman, J. Stat. Phys. 71, 89 (1993).[22] A.A. Lushnikov, Phys. Lett. 120A, 135 (1987); J.L.Spouge, Phys. Rev. Lett. 60 871 (1988); J.G. Amar andF. Family, Phys. Rev. A 41, 3258 (1990).[23] J.F.F. Mendes, R. Dickman, M. Henkel, and M.C. Mar-ques, J. Phys. A 27, 3019 (1994).4

Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer
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Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer Model

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