ofPercolationNetwork
HisaoNakanishi†
HLRZ,KFA–J¨ulich,Postfach1913
W-5170J¨ulich,Germany
arXiv:cond-mat/9301001v1 1 Jan 1993†
Presentandpermanentaddress:DepartmentofPhysics,PurdueUniversity,
WestLafayette,Indiana47907U.S.A.
Westudytheeffectsofaddingloopstoacriticalpercolationclusteronthedif-fusional,andequivalently,(scalar)elasticpropertiesofthefractalnetwork.Fromthenumericalcalculationsoftheeigenspectrumofthetransitionprobabilitymatrix,wefindthatthespectraldimensiondsandthewalkdimensiondwchangesuddenlyassoonasthefloppyendsofacriticalpercolationclusterareconnectedtogethertoformrelativelylargeloops,andthattheadditionalinclusionofsuccessivelysmallerloopsonlychangetheseexponentslittleifatall.Thissuggeststhatthereisanewuniversalityclassassociatedwiththeloop-enhancedpercolationproblem.
Keywords:Percolation,Diffusion,Vibration,Spectrum,Aerogel
PACSnumbers:64.60A,82.70G,05.40
1
I.Introduction
Oneofthefewexperimentallyaccessiblemeasuresofthedegreetowhichastruc-tureisfractal[1,2]isthespectraldimensionds[3].Thespectraldimensionwasoriginallydefined[4]fromthelowenergybehaviorofthevibrationaldensityofstatesn(E)ofafractal,elasticnetwork:
n(E)∼Eds/2−1,
(1)
whereEistheenergyofamode.Theenergyisproportionaltoω2whereωistheangularfrequencyofthevibrationalmode,andwritingtheaboveasymptoticrelationintermsofthedensityofstatesperunitintervalinω,wehavetherelation
ρ(ω)∼ωds−1,
(2)
inwhichdshasreplacedtheEuclideandimensiondinamorefamiliarequationforthephonondensityofstates.
Theexponentdscanbemeasuredbyvariousopticalandneutronscatteringex-periments[5],andforsomesilicaaerogelsinparticular,avalueintheneighborhood1.3±0.01wasfound.Now,suchafractalissometimescomparedtothecriticalpercolationcluster[6]sincethelatteristhesimplestandbeststudiedmodelofanequilibriumrandomfractal.Ifthiscomparisonismadeforthesilicaaerogel,theexperimentalvalueofdshappenstobeclosetothedsofthescalarelasticproblem[4]onthecriticalpercolationclusterind=3,andmuchlargerthanthecorrespondingvectorelasticresultofaboutds=0.9[1].(Vectorelasticityistheproblemwherethedisplacementofeachnodeinthenetworkisavectorratherthanascalar.)Courtens[5]arguedthatthisisnotanevidenceforscalarelasticitybutratherthatanaerogelhasveryfewfloppyendingsunlikeapercolationcluster.Thatis,mostfloppyendingswhichmayformduringthegrowthofanaerogelareconnectedtogethertoformloopsduringtheagingand(supercritical)dryingprocesses.Thusthefinalstructureismuchmorerigidwithhighervalueofdsalthoughthefractaldimensiondfisessentiallythesame.
Inthispaper,wetestthissuggestion,butforthesimpler,scalarelasticityproblem,ontwo-andthree-dimensionalcriticalpercolationclusters.Inthiscase,eachbond
2
connectingtwo(occupied)nearestneighborsitesoftheclusteractslikeanidealspringofuniformspringconstantinthesensethatitexertsaHooke’slawforcebasedonthedifferenceinthe(scalar)displacementsofthetwosites.Theeffectsofloopsarestudiedbyaddingcontrolledamount(andsize)ofloopsandcalculatingthedynamicexponentsdsanddw[3].Asfarasweknow,thisisthefirstquantitativestudyoftheeffectsofaddingloopsonthestructuralpropertiesofastochasticfractal.
Westudythisproblemasadiffusionproblemusingthevibration–diffusionanal-ogy[4]forscalarelasticity.Accordingtothiscorrespondence,e.g.,thevibrationalspectrumofthenetworkcanbecalculatedasthespectrumofthetransitionprobabil-itymatrixW[7,8],wheretheelementsWijofWissimplythehoppingprobabilityperstepfromnodejtonodeiinthediscretenetwork.ThusthismatrixWcontainstheinformationonboththenetworktopologyandthedynamicsofthemodelofdif-fusion.Wechooseforourcalculationstheso-calledblindantrandomwalk[3]asamodelofdiffusionalthoughthespecificchoiceofthetypeofrandomwalkisirrelevantforourresults.Forablindant,therandomwalkerattemptstomovewithouttheknowledgeofwhichneighborsarepartofthecluster(thusavailable)andwhichonesarenot,andthuswhenithappenstochooseanunavailableneighbor,itcannothopandmustwaitatitscurrentpositionforthenexttimestep.
Fromthepointofviewofthediffusionproblem,theinterestingquantitiesincludetheprobabilityP(t)forarandomwalkertoreturntoitsstartingpointaftertstepsandtheroot-mean-squaredistanceR(t)traveledbytherandomwalkintimet.Thesequantitiesareexpectedtohavethepower-lawsforasymptoticallylongtimest:
P(t)∼t−ds/2R(t)∼t1/dw.
Theserelationsmaybeconsideredtodefinetheexponentsdsanddw.
WecalculatetheseexponentsbyapproximatelydiagonalizingthematrixWusingthemethodofRef.[9,8].Oncethediagonalizationisdone,wecomputetwoquan-tities,thedensityofeigenvaluesn(λ)(whereλdenotestheeigenvaluesofW)andacertainfunctionπ(λ)(whichistheproductofn(λ)andsomecoefficientdeterminedwhenthestationaryinitialstatedistributionisexpandedintermsoftheeigenvectorsofW[10]).Thesefunctionsareexpectedtobehave,asymptoticallynearλ=1[8],
3
(3)(4)
as
n(λ)∼|lnλ|ds/2−1π(λ)∼|lnλ|1−2/dw.
II.LoopAddition
Inthissection,wecharacterizetheclustersforwhichthecalculationsdescribedinSectionIareperformed.Theprocedureisasfollows:wefirstgeneraterandomrealizationsofoccupiedsiteswiththeknowncriticalprobabilitypc(independentlyofeachother)oneitherasquareorsimple-cubicgridofapredeterminedsize(edgelengthL).Thepercolationproblemisthenconstructedbyimposingthenearest-neighborconnectivityamongtheoccupiedsitesusingperiodicboundaryconditioninalldirections.Wethenchooseonlythoserealizationswhosemaximalclusterspansthegridinalldirectionsaswellaswrapsinalldirections,i.e.,thatthoseonwhichthereisapaththatwindsaroundthegrid(withtheperiodicboundaries)ineachcoordinatedirection.Themaximal,wrappingclustergeneratedinthiswayisanallowablestartingconfiguration.
Inordertoaddloopsinacontrolledmanner,wefirstmarkallperimetersiteswhichareneighborstotwoormoreoccupiedsites(i.e.,thoseemptysiteswhichwouldconnecttwoormoreoccupiedsitesiftheywereoccupied).Foreachofthesemultiple-perimetersites,wemustdecidethesizeoftheloopitcloses.Whileitisveryeasytodeterminethesizeofthesmallestloop,thatisnotaveryinterestingquantity,sinceitmayalsocloseamuchlargerloopandsuchalargerloopmaydominatetherigidityofthestructure.Ontheotherhand,thesizeofthelargestloopitselfisalsonotaninterestingquantitysinceinprincipletherearemanyloopsthatareembeddedintheinterioroftheclusterwithlittleinfluenceonthecluster’srigidity.
Thusweusethefollowingproceduretodeterminetheeffectivesizeoftheloopamultiple-perimetersitecloses:Foreachsuchsite,weconsidereverypairofoccupiedneighborsinturn.Foreachpair,wecalculatetheminimumloopsizebyaburningmethod[11],spreadingafirefromeachendanddeterminingthefirstcontactofthetwofires.Thenwecomputethemaximumamongtheseminimumloopssizesattachedtotheperimetersiteinquestion.ThisistheappropriateloopsizeP,thesizeofthe
4
(5)(6)
floppiestloopassociatedwiththeparticularmultiple-perimetersite.Wethenaddallmultiple-perimetersiteswithPgreaterthanorequaltoacertainpredeterminedvaluePotocontrolhowfloppyaloopmustbeforittobeclosed.Forexample,Po=∞willclosenoloop,andP=4willcloseallloopsforboththesquareandsimplecubiclattices.
Inthispaper,wecomparetheresultsforPo=∞,12,8,and6.Atypicalstartingclusterandthecorrespondingloop-addedclustersonthesquarelatticeareshowninFig.1.Clearly,thefuzzyandfloppyendingspresentintheoriginalcluster(a)becomethickerandbetterconnectedsuccessivelyin(b),(c),and(d)asmoreloopsareadded,givingtheimpressionofimagesharpening.Thefinalstructurein(d)withPo=6appearstobeamuchmoresolidobjectthantheoriginal;yet,thefractaldimensionofalltheseclustersarevirtuallythesame,beingabout1.9.
Theaboveprocedureisfollowedafterthestartingconfigurationisfixed.Theapplicablemultiple-perimetersitesareorderedinaparticularway(inatypewriterfashionduetotheconstructionofthecluster),andtheadditionprocedureappliedonlyonceinthefixedorder,withnorecursion.Thatis,onceasiteisdeterminedtobeeitheraddedornot,wedonotlaterconsideritagainforadditionandalsowedonotconsiderperimetersitesnewlycreatedbythisprocessitself.ThisremovestheproblemofthecompletefillingoflakesandfjordswhichwouldoccurparticularlywhenPoissmall,butdoesnotsolvetheproblemofthefinalstructuredependingsomewhatontheinitialorderingoftheperimetersites.Physically,norecursionrulemightcorrespondtotheassumptionthattheformationofadditionalbondsduringaginganddryingislargelysimultaneousandnotsequential.III.NumericalResults
Ournumericalresultsforn(λ)forthedensityofstatesandπ(λ)(relatedtothevelocityautocorrelationfunctionoftherandomwalkbyaLaplacetransform[9,8])areplottedinFig.2andFig.3respectively.Weseethatgenerallyd=3dataappeartobebehavinginaclearerwaythanthoseofd=2.WesummarizetheexponentestimatesfordsanddwinTablesIandIIwheretheerrorestimatesaremostlyfromtheleastsquaresregressionanddonottakeintoaccountanyfinitesizeeffectsorothersystematicerrorsthatmaybepresent.
5
ThegridsizesforthecalculationsshowninthefiguresareL=100forthesquarelatticeandL=30forthesimplecubiclattice(althoughwehavealsocheckedfortheclustersizedependenceusingsmallergridsofL=50forthesquarelatticeandL=20forthesimplecubiclattice).Thespanningandwrappingpercolationclusters(atp=0.593≈pcforthesquareandp=0.312forthesimplecubiclattice)containedabout4600sitesford=2and4000sitesford=3beforeaddinganyloops.ForPo=12,thesizeincreasestoabout5050and4400,forPo=8,toabout5250and4900,andforPo=6,to6100and5700ford=2andd=3,respectively.ForeachvalueofPo,averagesweretakenover400independentclusterrealizationsforthesquarelattice,andforthesimplecubiclattice,thenumberofclustersusedvariedslightlyas481,480,and479forPo=12,8,and6,respectively.
Thus,boththesizeoftheclustersandthenumberofindependentsamplesinthedisorderensemblearerathermodest;however,theyarecomparabletothepreviouscalculationsbythesamemethodforthePo=∞case[8]wheretheresultsforpcwereinexcellentagreementwithothercalculationsofdsanddwaswellaswiththescalingrelationds=2df/dw[3](butseealso[12]).ThescopeofthecalculationswereCPUtimelimitedmainlybecauseoftheneedtocalculatetheeigenvaluesandeigenvectorsveryaccurately(usuallyto6digits)forthelargest200orsoeigenvalues.Thesecalculationstooktypically3hoursofCPUtimeononeCrayY-MPprocessorforeachPoforthesquarelatticeand6hoursforeachPoforthesimplecubiclattice.PerformingthesecalculationsonworkstationsforlongperiodsoftimewouldsolvetheCPUlimitation,butthenthememorybecomesthelimitingfactor.
Sincetheslopeinthelog-logplotofFig.2mustequalds/2−1(cf.Eq.(5)),itisobviousthattheoriginalpercolationclusters(Po=∞)yieldcompletelydifferentvaluesofdsinbothd=2and3evenfromthecaseofPo=12.Indeed,theestimateofdsisabout1.30forbothd=2andd=3ifPo=∞,butfortheloop-addedcases,theestimatesareds≈1.7ford=2and≈2.0ford=3.SincethecasePo=12containsonlyabout10%moresitesandvisuallylooksverysimilartotheoriginalpercolationcluster(cf.Fig.1(a)and(b)),thisisaclearevidencethatthestructuralpropertiesareverysensitivetotheadditionofevenafewloops.Ontheotherhand,theslopesofthedatapointsforthethreefinitevaluesofPodonotdifferverymuchfromeachother.Thusoncethelargerloopsareadded,anyfurtheradditionofsmaller
6
loopsdonotappeartoaffecttheresultsverymuch.
Theredoesseemtobeatendencyfortheslopetogetsteeperwhenmoreloopsareadded.Thistendencyiswithinthestatisticalstandarddeviationsford=2butdefinitelyoutsidereasonablestatisticalerrorsford=3.Astudyofsmallerclusters(L=50forthesquareandL=20forthesimplecubiclattices)alsoshowsdifferentbehaviorsford=2and3.Forthesquarelattice,thisslopegetsconsistentlysteeperforallPo=12,8,and6asLincreases;however,theincreaseintheslopeiswellwithinthestatisticalstandarddeviationinallcases.Forthesimplecubiclattice,ontheotherhand,theslopebecomeslesssteepforPo=12andsteeperforPo=6asLincreases,thechangesbeingslightlyoutsideofthestatisticalerrors.TheseconsiderationssuggestthatthedifferentdsfordifferentPo<∞mightbearealeffectford=3butlesslikelytobesoford=2.However,amorecompletestudy,e.g.,ofthefinitesizeeffectsisneededtodefinitivelyanswerthisquestion.Inanycase,thedifferencesintheslopeamongthePo=12,8,and6arefarsmallerthanthedifferencebetweenthesevaluesofPoandPo=∞.
Comparedtotheresultsforthedensityofstatesn(λ),theresultsforπ(λ)inFig.3showasimilar,butmuchlessdrasticrelativedifferencesindwbetweentheloop-addedclustersandtheoriginalpercolationclusters.Thatis,theexponentdwchangesfromabout2.9fortheoriginalclustertoabout2.4fortheloop-addedcaseford=2,achangeofonlyabout17%comparedtothecaseofdswherethechangeismorethan30%.Thiscanbeunderstoodsincetheadditionofafewloopsmaynotchangethevelocityautocorrelationofarandomwalkerintstepsasmuchasitschancestoreturntothestartingpoint,orputanotherway,thestructuralrigidityismuchmoresensitivetotheadditionofafewloopsthantheoverallrandomwalkdisplacementonthesamenetwork.
Again,thethree-dimensionalresultsappeartobebetterbehavingthanthetwo-dimensionalones.However,inbothcases,theslopesinthelog-logplotofFig.3aresignificantlysteeperforPo=∞thanfortheloop-addedclusters,withahintofaslighttrendforthelargerslopeforPo=6ind=3.Sincethisslopemustequal1−2/dw(cf.Eq.(6)),alargerslopeimpliesalargervalueofdw.ThesedifferencesamongPo<∞are,however,atleastasill-definedasforthecaseofn(λ).
7
IV.Summary
Insummary,wehavepresentedtheanalysisofthedynamicalpropertiesoftherandomwalkconfinedtoloop-enhancedcriticalpercolationclusters.Throughthevibration–diffusionanalogy[4],thisworkhasimplicationsforthe(scalar)elastic-ityproblemonthesamefractalnetwork.Fromthenumericalcalculationsoftheeigenspectrumofthetransitionprobabilitymatrix,wefindwhatappearstobethecrossovertoanewuniversalityclasscharacterizedbyasignificantlylargerspectraldimensiondsandsignificantlysmallerwalkdimensiondwassoonasthefloppyendsofacriticalpercolationclusterareconnectedtogethertoformrelativelylargeloops.Theadditionalinclusionofsuccessivelysmallerloopsdoesnotseemtofurtherchangetheexponentsdsanddwverymuch.ThisresultsupportstheobservationofRef.[5]thatthedeviationbetweentheexperimentalobservationofdsofanaerogelandtheprevioustheoreticalcalculationsforthevectorelasticityofthecriticalpercolationclusterisduetothelackoffloppyendingsintheaerogelbecauseofthechemicalreactionsduringagingand(supercritical)drying.
Theloop-enhancedpercolationproblemisreminiscentoftheproblemoftheex-ternalsurfaceofthecriticalpercolationcluster.Thelattercanbedefinedinseveralways,e.g.,bythepercolationhull[6]andbytheperimetersitesaccessibletoarandomwalkfromtheoutside.Theaccessibilityinthatcasedependsonthesizeofthewalker[13];inasomewhatsimilarway,thefloppinessoftheloopsinourproblemdependsontheminimumloopsizePo.Asinourcase,theexternalperimeterproblemalsoleadstoonenewuniversalityclass(inadditiontotheoneforthehull)intwodimensions[13].Onedifferenceis,however,thatourproblemleadstoanewuniversalityclassalsointhreedimensionswhilethisisnotsofortheexternalperimeterproblem.Beyondthequalitativeanalogy,theremightlurkamorequantitativeone,sincetheproblemoftheexternalsurfaceisthatofsuccessivelycontrollingtheaccessibilityforaparticlediffusingfromoutsidewhileourproblemisthatofsuccessivelycontrollingthereachofaparticlediffusingontheclusteritself.Thisis,however,beyondthescopeofthisworkandmustawaitfurtherresearch.
8
Acknowledgments
IamgratefultoHansHerrmannforsuggestingthisproblemandtohimandDietrichStaufferforhelpfuldiscussions.ThisworkwascarriedoutwhileIwasvisitingtheHLRZSupercomputerCenteratKFA–J¨ulichinGermany.IwouldliketothanktheCenterandHansHerrmannforwarmhospitality.
9
References
[1]B.Mandelbrot,FractalGeometryofNature(Freeman,SanFrancisco,1982).[2]J.Feder,Fractals(Plenum,NewYork,1988).
[3]See,e.g.,S.HavlinandD.Ben-Avraham,Adv.Phys.36(1987)695.[4]S.AlexanderandR.Orbach,J.Phys.Lett.(Paris)43(1982)625.
[5]E.Courtens,R.VacherandE.Stoll,inFractalsinPhysics,eds.A.Aharonyand
J.Feder(North-Holland,Amsterdam,1989).
[6]D.StaufferandA.Aharony,IntroductiontoPercolationTheory(Taylorand
Francis,London,1992).
[7]A.B.Harris,Y.MeirandA.Aharony,Phys.Rev.B36(1987)8752.[8]H.Nakanishi,S.MuckherjeeandN.H.Fuchs,preprint(1992).[9]N.H.FuchsandH.Nakanishi,Phys.Rev.A43(1991)1721.[10]D.JacobsandH.Nakanishi,Phys.Rev.A41(1990)706.
[11]H.J.Herrmann,D.C.HongandH.E.Stanley,J.Phys.A17(1984)L261.[12]H.NakanishiandH.J.Herrmann,HLRZpreprint.
[13]T.GrossmanandA.Aharony,J.Phys.A16(1986)L745;20(1987)L1193.
10
FIGURES
Fig.1:Theresultofaddingsuccessivelysmallerloopstothecriticalpercolation
clusterisillustratedwith:(a)atypicalspanningandwrappingcriticalpercola-tionclusteronthesquarelattice;(b)whenallloopsofloopparameterP≥12areadded;(c)whenallloopsofP≥8areadded;(d)whenallloopsofP≥6areadded.Thegridsizeis100×100.
Fig.2:Densityofeigenvaluesn(λ)for(a)thesquarelatticeand(b)thesimple
cubiclattice.Thesymbols×,,2,and∆correspondtoPo=∞,12,8,and6,respectively.Linesdrawnarelinearleastsquaresfitstothecorrespondingdatashownhere.ThedataforPo=∞arefromRef.[8].
Fig.3:Thefunctionπ(λ)for(a)thesquarelatticeand(b)thesimplecubiclattice.
Thesymbols×,,2,and∆correspondtoPo=∞,12,8,and6,respectively.Linesdrawnarelinearleastsquaresfitstothecorrespondingdatashownhere,andthedataforPo=∞arefromRef.[8]
11
TABLES
Table1:Numericalestimatesoftheexponentsdsanddwforloop-enhancedcriticalpercolationclusteronthesquarelattice.TheresultsforPo=∞arefromRef.8.Theerrorestimatesaremainlyfromtheleastsquaresregression.Po
ds
dw
∞1.30±0.023.70±0.07121.99±0.022.67±0.0781.94±0.022.76±0.086
1.89±0.012.76±0.04
12
因篇幅问题不能全部显示，请点此查看更多更全内容