Nonequilibrium dynamics of urn models

arXiv:cond-mat/0109213v1[cond-mat.stat-mech]12Sep2001NonequilibriumdynamicsofurnmodelsCGodr`eche†‡andJMLuck¶§†ServicedePhysiquedel’´EtatCondens´e,CEASaclay,91191Gif-sur-Yvettecedex,France¶ServicedePhysiqueTh´eoriquek,CEASaclay,91191Gif-sur-Yvettecedex,FranceAbstract.Dynamicalurnmodels,suchastheEhrenfestmodel,haveplayedanimportantroleintheearlydaysofstatisticalmechanics.Dynamicalmany-urnmodelsgeneralizetheformermodelsintworespects:thenumberofurnsismacroscopic,andthermaleffectsareincluded.Thesemany-urnmodelsareexactlysolvableinthemean-fieldgeometry.Theyallowanalyticalinvestigationsofthecharacteristicfeaturesofnonequilibriumdynamicsreferredtoasaging,includingthescalingofcorrelationandresponsefunctionsinthetwo-timeplaneandtheviolationofthefluctuation-dissipationtheorem.Thisreviewpapercontainsageneralpresentationofthesemodels,aswellasamoredetaileddescriptionoftwodynamicalurnmodels,thebackgammonmodelandthezetaurnmodel.PrologueUrnscontainingballs,justasdiceorplayingcards,areubiquitousinwritingsonprobabilitytheory,remindingusthatthisbranchofmathematicsowesitsearlydevelopmentstopracticalquestionsarisinginplayinggames.Dynamicalurnmodels,suchastheEhrenfestmodel,haveplayedanimportantroleintheelucidationofconceptualproblemsinstatisticalmechanics.Morerecently,dynamicalmany-urnmodelshavebeeninvestigatedinthemean-fieldgeometry.Theseexactlysolvablemodelsexhibitcharacteristicfeaturesofnonequilibriumdynamicsreferredtoasaging,suchasthescalingofthecorrelationandresponsefunctionsinthetwo-timeplaneandtheviolationofthefluctuation-dissipationtheorem.Thispapercontainsadidacticintroductiontournmodels(Section1),apresenta-tionofstaticanddynamicalpropertiesofmany-urnmodels(Section2),areminderofthemaincharacteristicfeaturesofaging(Section3),andanoverviewofrecentresultsontwodynamicalurnmodels,namelythebackgammonmodel(Section4)andthezetaurnmodel(Section5).1.Urnmodels1.1.TheEhrenfesturnmodelThismodelwasdevisedbyPandTEhrenfestin1907[1]intheirattempttocriticallyreviewBoltzmann’sH−theorem.Itisdefinedasfollows:Nballsaredistributedin‡[e-mail]§[e-mail]2306ofCNRSNonequilibriumdynamicsofurnmodels2twourns(orboxes).Atrandomtimes,givenbyaPoissonprocesswithunitrate,aballischosenatrandom,andmovedfromtheboxinwhichitistotheotherbox.LetN1(t)(resp.N2(t)=N−N1(t))bethenumberofballsinboxnumber1(resp.number2)attimet.Foranyinitialconfiguration,thesystemrelaxestoequilibriumforinfinitelylongtimes.TheequilibriumstateischaracterizedbyabinomialdistributionofthenumberN1:fk,eq=Peq(N1=k)=2−NNk(k=0,...,N),(1.1)reminiscentoftheMaxwell-Boltzmannstatisticsforindistinguishableparticles.AsnoticedfirstbyKohlrauschandSchr¨odinger[2],N1(t)canbeviewedastheco-ordinateofafictitiouswalker.Thetemporalevolutionoftheoccupationprobabilitiesfk(t)=P(N1(t)=k)isgovernedbythemasterequationdfk(t)dt=k+1Nfk+1(t)+N+1−kNfk−1(t)−fk(t).(1.2)Indeed,amoveofaballfromboxnumber1toboxnumber2(resp.fromboxnumber2toboxnumber1)yieldsk→k−1(resp.k→k+1),andoccurswitharatek/N(resp.(N−k)/N)perunittime.Themasterequation(1.2)describesanon-uniformlybiasedrandomwalkovertheintegers0,...,N[2,3].Thefirst(resp.thesecond)termintheright-handsideisabsentfork=N(resp.k=0).Thespectrumofrelaxationratesofthemodelcanbederivedbylookingforsolutionsto(1.2)oftheformfk(t)=φke−λt.Onethusobtainsλm=2m/N,withm=0,...,N.Thestaticsolution(m=0)isnothingbuttheequilibriumdistribution(1.1).Theoccupationprobabilitieshaveanexponentialconvergencetotheirequilibriumvalues,witharelaxationtimeτeq=1/λ1=N/2.Thereare,however,muchlargertimescalesintheEhrenfestmodel.Considerindeedthelengthoftimeittakesforboxnumber1togetempty,i.e.,thefirsttimet0suchthatN1(t0)=0.Thistimedependsontheinitialstateandonthewholehistoryofthesystem.ItsmeanvalueT0=ht0icanbeevaluatedasfollows.Theequilibriumprobabilityforboxnumber1tobeempty,f0,eq=2−N,isexponentiallysmallinthenumberofballs.ItisthereforeexpectedthatthetypicaltimeneededtoreachthisveryrareeventscalesasT0≈1/f0,eq=2N.(1.3)Thisresultcanbederivedinamorerigorousway.RelatedmattersarediscussedinReferences[4,5].TheprocessofemptyingoneoftheboxesisthereforecharacterizedbyanexponentiallylargetimeT0.Equation(1.3)canberecastasanArrhenius-likelaw:T0∼eS0,(1.4)whereS0=Nln2istheentropydifferencebetweentheequilibriumstateofthemodel,whereeachboxcontainsonehalfoftheballs,uptofluctuations,andtheconfigurationwhereboxnumber1isempty.Thisentropydifferenceplaystheroleofareducedactivationenergyin(1.4).Thisisanelementaryexampleofanentropybarrier.Nonequilibriumdynamicsofurnmodels31.2.TheMonkeyurnmodelLetusintroduceavariantoftheEhrenfesturnmodel,whichwechoosetocalltheMonkeyurnmodel,becauseitcorrespondstotheimageofamonkeyplayingatexchangingballsbetweentwoboxes.Thekeydifferencewiththepreviouscaseisthatnowateachtimestepabox(either1or2)ischosenatrandom,andoneballismovedfromthechosenbox(provideditisnon-empty)totheotherone.Thechoiceofabox,insteadofaball,inducesdrasticchangesinthestaticsanddynamicsofthemodel.TheequilibriumstateoftheMonkeyurnmodelisnowcharacterizedbyuniformoccupationprobabilitiesfk,eq=Peq(N1=k)=1N+1(k=0,...,N).(1.5)Theev