Phase diagram of a probabilistic cellular automato

arXiv:cond-mat/0210036v1[cond-mat.stat-mech]1Oct2002Phasediagramofaprobabilisticcellularautomatonwiththree-siteinteractionsA.P.F.Atman∗,RonaldDickman†andJ.G.Moreira‡DepartamentodeF´ısica,InstitutodeCiˆenciasExatas,UniversidadeFederaldeMinasGerais,C.P.70230123-970,BeloHorizonte,MG-Brazil(February1,2008)AbstractWestudya(1+1)dimensionalprobabilisticcellularautomatonthatiscloselyrelatedtotheDomany-Kinzel(DKCA),butinwhichtheupdateofagivensitedependsonthestateofthreesitesattheprevioustimestep.Thus,comparedwiththeDKCA,thereisanadditionalparameter,p3,representingtheprobabilityforasitetobeactiveattimet,giventhatitsnearestneighborsanditselfwereactiveattimet−1.Westudyphasetransitionsandcriticalbehaviorfortheactivityandfordamagespreading,usingone-andtwo-sitemean-fieldapproximations,andsimulations,forp3=0andp3=1.Wefindevidenceforalineoftricriticalpointsinthe(p1,p2,p3)parameterspace,obtainedusingamean-fieldapproximationatpairlevel.Toconstructthephasediagraminsimulationsweemploythegrowth-exponentmethodinaninterfacerepresentation.Forp3=0,thephasediagramissimilartotheDKCA,butthedam-agespreadingtransitionexhibitsareentrantphase.Forp3=1,thegrowth-exponentmethodreproducesthetwoabsorbingstates,firstandsecond-orderphasetransitions,bicriticalpoint,anddam-agespreadingtransitionrecentlyidentifiedbyBagnolietal.[Phys.Rev.E63,046116(2001)].PACS:05.10.-a,02.50.-r,68.35.Ct,68.35.Rh∗atman@fisica.ufmg.br†dickman@fisica.ufmg.br‡jmoreira@fisica.ufmg.br1I.INTRODUCTIONProbabilisticcellularautomata(PCA)arewidelyusedtomodelsystemswithlocalinteractionsinphysics,chemistry,biologyandsocialsciences[1,2,3,4,5].Despitetheirsimplicity,thesemodelsexhibitcomplexbehaviorandareusedtoinvestigatefundamentalproblemsinstatisticalmechanics,suchasspinmodels[6,7]andnonequilibriumphenomena[8,9].Inparticular,theproblemofphasetransitionsinthepresenceofabsorbingstateshasattractedincreasinginterestinrecentyears;PCAplayamajorroleinthesestudies[10,11,12,13].ThePCAintroducedbyDomanyandKinzel[8]is,alongwiththecontactprocess[14,15],oneofthesimplestmodelsexhibitinganabsorbing-statephasetransition.Theone-dimensionalDomany-Kinzelstochasticcellularautomaton(DKCA)isacompletelydiscretesystem-temporally,spatiallyandinitsstatespace-whichattractsinterestasaparticlesystemaffordingatestofideasonscalinginnonequilibriumcriticalphenomena[16].TheDKCAhasauniqueabsorbing(“vacuum”)state;itsphasediagrampresentsacriticallineseparatingthisab-sorbingphasefromanactiveone.Continuousphasetransitionstoanabsorbingstateareconjecturedtobelonggenericallytothedirectedpercolation(DP)uni-versalityclass[17].Inadditiontotheactive-absorbingtransition,Martinsetal.[18]foundadamagespreading(DS)transitionseparatingtheactivephaseintononchaoticandchaoticphases.Thereisnumericalevidencethatthecriticalbe-havioralongthistransitionlinealsobelongstotheDPclass,asexpectedonthebasisofuniversality[19].Recently,Bagnolietal[9]introducedamodelthatcanbeconsideredanaturalextensionoftheDKCA:aone-dimensionalPCAinwhichtheupdateofagivensitedependsonthestateofitsnearestneighborsanditself,attheprecedingtimestep.(WeshallrefertothismodelastheBPCA.)Thus,comparedwiththeDKCA,thereisanadditionalparameter,p3,representingtheprobabilityforasitetobeactiveattimet,giventhatallthreesiteswereactiveattimet−1.Bagnolietal.studiedp3=1,inwhichcasethemodelpresentstwoabsorbingstates:theemptyoneandthecompletelyoccupiedconfiguration.AsintheDKCA,thedensityistheorderparameter.Theseauthorsusedthemeanfieldapproximation(atsitelevel),simulations,andfield-theoreticargumentstostudythemodel.Theyfoundarichphasediagram,withfirst-andsecond-orderphasetransitions,abicriticalpoint,andadamage-spreadingtransition.Exceptforthelinep2=1intheDKCA,thisisthesimplestPCAthatexhibitsadiscontinuousphasetransition[9].InthisworkweextendtheanalysisoftheBPCAconsideringtwocases:thepreviouslystudiedp3=1,whichcorrespondstoaferromagnetic-likemodel,andp3=0,representingagame-of-life-likemodel[20,21].Weextendthemean-fieldanalysistothepairlevel,andusesimulationstoconstructthephasediagram.Insimulations,weapplythegrowth-exponentmethod[22]toidentifytransitions.Thispaperisstructuredasfollows:inSectionIIwedefinethemodelanditsinterfacerepresentation;thesiteandpairmean-fieldapproximationsarediscussedinSectionIII.SimulationresultsarepresentedinSectionIV.Wesummarizeour2findingsinSectionV.II.MODELTheone-dimensionalPCAwiththree-siteneighborhood(BPCA)waspro-posedbyBagnolietal.[9].ItconsistsofaringofLsites(i=1,2,...,L),withperiodicboundaries,inwhicheachsiteihastwopossiblestates,conve-nientlydenotedbyσi=0,1.Thestateofthesystemattimetisgivenbytheset{σi(t)}.IncontrasttothedeterministicCAstudiedbyWolfram[1],thepresentmodelisadiscretetimeMarkovprocess:therulesforupdatingthesystemaregivenbytransitionprobabilities.Inparticular,stateofsiteiattimet+1dependsonσi−1(t),σi(t)andσi+1(t),viathetransitionprobabilityP(σi(t+1)|σi−1(t),σi(t),σi+1(t)).Thelatterisoftotalisticform,i.e.,thedepen-denceisthroughSi(t)=σi−1(t)+σi(t)+σi+1(t).SinceS(t)=0impliesσi(t+1)=0withprobability1,thereremainthreefreeparametersfordefiningthetransitionprobability.Sp