The Asymmetric Simple Exclusion Process with Multi

arXiv:math/9911237v1[math.PR]29Nov1999THEASYMMETRICSIMPLEEXCLUSIONPROCESSWITHMULTIPLESHOCKSPabloA.Ferrari(1),L.RenatoG.Fontes(1),M.Eul´aliaVares(2)Abstract:Weconsidertheonedimensionaltotallyasymmetricsimpleexclusionprocesswithinitialproductdistributionwithdensities0≤ρ0ρ1...ρn≤1in(−∞,c1ε−1),[c1ε−1,c2ε−1),...,[cnε−1,+∞),respectively.Theinitialdistribu-tionhasshocks(discontinuities)atε−1ck,k=1,...,nandweassumethatinthecorrespondingmacroscopicBurgersequationthenshocksmeetinr∗attimet∗.Themicroscopicpositionoftheshocksisrepresentedbysecondclassparticleswhosedistributioninthescaleε−1/2isshowntoconvergetoafunctionofninde-pendentGaussianrandomvariablesrepresentingthefluctuationsoftheseparticles“justbeforethemeeting”.Weshowthatthedensityfieldattimeε−1t∗,inthescaleε−1/2andasseenfromε−1r∗convergesweaklytoarandommeasurewithpiecewiseconstantdensityasε→0;thepointsofdiscontinuitydependontheselimitingGaussianvariables.Asacorollaryweshowthat,asε→0,thedistri-butionoftheprocessatsiteε−1r∗+ε−1/2aattimeε−1t∗tendstoanontrivialconvexcombinationoftheproductmeasureswithdensitiesρk,theweightsofthecombinationbeingexplicitlycomputable.KeywordsAsymmetricsimpleexclusionprocess,dynamicalphasetransition,shockfluctuationsAMSsubjectclassification:60K35,82CPartiallySupportedbyCNPqandFAPESPandFINEP(PRONEX).(1)IME-USP.CaixaPostal66.281,S˜aoPaulo05315-970SPBrasil(2)IMPA.EstradaD.Castorina110,J.Botˆanico22460-320RJBrasil12§1.Introductionandresults.Itiswellknownthatthehydrodynamicalbehavioroftheonedimensionalasym-metricexclusionprocessisdescribedbytheinviscidBurgersequation(1.1)∂tρ+γ∂r(ρ(1−ρ))=0whereγisthemeanofthejumpdistribution.Sinceequation(1.1)developsdiscon-tinuities,onehastobecarefulabouttheprecisestatement,butlooselyspeaking,if(r,t)isacontinuitypointofρ(r,t),foragiveninitialmeasurableprofileρ0(·),thenatthemacroscopicpoint(r,t)thesystemisdistributedaccordingtothemeasureνρ(r,t),whereνρistheproductBernoullimeasureon{0,1}Zwithνρ(η(x)=1)=ρforallx.Thisisknownaslocalequilibrium.Asitisknown,theexactstatementinvolvesaspace/timechange,underEulerscale,andforallthesedevelopmentswerefertoAndjelandVares(1987),Rezakhanlou(1990),Landim(1992).Theproblemwithwhichweareconcernedhereinvolvesthedescriptionofthe(microscopic)behaviorofthesystematcertaindiscontinuitypoints(r,t)(orshockfronts)ofthesolutionofequation(1.1).Forexample,ifγ0andtheinitialprofileisastepfunctionρ0(r):=α1{r0}+β1{r≥0},with0≤αβ≤1,theentropysolutionofequation(1.1)isρ(r,t)=α1{rvt}+β1{r≥vt},wherev:=γ(1−α−β)isthevelocityoftheshockfrontand1{·}istheindicatorfunctionoftheset{·}.Thisdescriptionisvalidonlyforcontinuitypoints.TheinvestigationofwhathappenstothesystemiflookedfromthisshockfrontwasfirststudiedbyWick(1985)foradifferentmodel,andfortheasymmetricsimpleexclusionintheparticularsituationsα=0andα+β=1byDeMasietal(1988)andAndjel,BramsonandLiggett(1988),respectively.Theyallprovedthatattheshockfrontoneseesafairmixtureofναandνβ.Thisresulthasthenbeenextendedsoastocoverallcases0≤αβ≤1byFerrariandFontes(1994),fromnowonreferredas[FF].[FF]workedwiththenearestneighborasymmetricexclusionprocess,whosegeneratoristheclosureof(1.2)Lf(η):=Xx∈ZXy=x±1p(x,y)η(x)(1−η(y))(f(ηx,y)−f(η))3forfacylinderfunctionin{0,1}Z,withηx,y(z):=η(z)ifz6=x,yη(y)ifz=xη(x)ifz=y,wherep(x,x+1):=p,p(x,x−1):=q:=1−p,with1/2p≤1.ThisprocesswasfirststudiedbySpitzer(1970).Callingμα,βtheproductmeasureon{0,1}Zwithsitemarginals(1.3)μα,β(η(x)=1):=αifx0βifx≥0,denotingStasthesemigroupcorrespondingtotheabovegeneratorandθxasthespaceshiftμθx(f):=Rf(θxη)μ(dη),withθxη(z):=η(x+z),[FF]provedthat(1.4)μα,βStθ[vt]ω∗−→12(να+νβ)ast→+∞,andwhere[x]denotestheintegerpartofx.Thiscorrespondstotheexactstatementmadeabove,underEulerscale,andforthemacroscopicpoint(r,t)inthefrontline,i.e.r=vt.Infact,intheabovementionedreferences,moredetailedanalysisisperformed,bylookingatthemicroscopicstructureoftheshockrepresentedbyasecondclassparticle.Asecondclassparticlejumpstoemptysiteswiththesameratesastheotherparticles,butinterchangespositionswiththeregularparticlesattherateholesdo.Aformaldefinitionusingcouplingisgiveninthenextsection.CallingXtthepositionofasecondclassparticleaddedattheorigin,theprocessasseenfromthesecondclassparticleθXtηthasdistributionasymptoticallyproducttotherightandleftoftheoriginwithdensitiesαandβrespectively,uniformlyintime.ThevelocityofthesecondclassparticleisthesameasthevelocityoftheshockintheBurgersequation:Eμα,βXt=γ(1−α−β)t.[FF]provedthatthefluctuationsofthepositionoftheparticleareGaussian:calling˜Xt:=Xt−vt,(1.5)˜Xt/√tD−→t→+∞Nα,β4whereNα,βisacenteredGaussianrandomvariablewithvarianceγ(β−α)−1(α(1−α)+β(1−β)).Withthisresultinhand[FF]provedthat,if−∞a+∞,thedistributionoftheprocessattimetatthepointvt+a√tconvergestoamixtureofναandνβ;moreprecisely,forreala,(1.6)μα,βStθ[vt+a√t]ω∗−→t→+∞ναP(Nα,βa)+νβP(Nα,β≤a)whichinparticularyields12(να+νβ)fora=0andinterpolatesbetweenναandνβ,asavariesfrom−∞to+∞.Ourgoalistheconsiderationoftwoormoreshockfrontsandthedescriptionofthemicroscopicbehaviorofthesystematthe(macroscopic)timeandpositionoftheirmeeting.Toavoidunnecessarytechni