Critical dimensions of the diffusion equation

(1)

T.J.Newman(1)+andWillLoinaz(2)∗

DepartmentsofPhysicsandBiology,UniversityofVirginia,Charlottesville,VA22904

(2)

DepartmentofPhysics,AmherstCollege,Amherst,MA01002

arXiv:cond-mat/0009365v1 [cond-mat.stat-mech] 22 Sep 2000Westudytheevolutionofarandominitialfieldunderpurediffusioninvariousspacedimensions.Fromnumericalcalculationswefindthatthepersistencepropertiesofthesystemshowsharptran-sitionsatcriticaldimensionsd1≃26andd2≃46.Wealsogiverefinedmeasurementsofthepersistenceexponentsforlowdimensions.PACSnumbers:05.40.-a

DiffusionisaubiquitousprocessinNatureandisoffundamentalimportanceinphysical,chemical,andbi-ologicalsystems.Purediffusion,asdescribedbythesimpleheatequation,hasbeensowell-studiedoversomanyyearsthatitisdifficulttoimaginethereisstillmuchaboutittobeunderstood.Recently,thepersis-tenceprobability,q(t),ofthediffusionequationinone,two,andthreedimensionswasmeasuredandfoundtobedescribedbynon-trivialpowerlaws[1]:q(t)∼t−θ(d),wheredisthedimensionofspace.Asyetthereisnoexactpredictionfortheexponentsθ(d).

Thestudyofpersistenceisarapidlyemergingfield[2],thepurposeofwhichistoprobethestatisticalnatureofdynamicalprocesseswithmoreresolutionthanthatallowedbyconventionalmeasures,suchasautocorrela-tionfunctions.Ithasbeenappliedtoawiderangeofsystemsincludingphaseordering,interfacekinetics,andthediffusionequation.Directexperimentalobservationofpersistencehasbeenachievedinthephaseorderingofliquidcrystals[3]andjustrecentlybyNMRstudiesofthediffusingpolarizationofaonedimensionalgas[4].Strictlyspeaking,thepersistenceprobabilitymeasurestheprobabilitythataparticulareventhasnotoccurreduptotimet.Moreloosely,by“persistentproperties”,weincludeotherstatisticalmeasures,suchasthesign-timedistribution(STD)[5,6].TheSTDisahistogrammeasuringthefractionoftimetabimodalvariablehasbeeninoneofitstwostates.ExamplesarethefractionoftimeanIsingspinis“up”,orthefractionoftimeastochasticvariableisaboveitsmean.ThetailsoftheSTDimplicitlyencodethepersistenceprobability.TheshapeoftheSTDyieldsmuchusefulinformationaboutthespatio-temporalstructureofthedynamicalprocess.Forexample,itcanbeusedtoinferlong-livedfeaturesininterfacesundergoingkineticroughening[7].InthisLetterweintroduceasimplealgorithmwhichallowsustostudythepersistencepropertiesofthediffu-sionequationinarbitrarydimensions.OurmainresultisthatthereexisttwocriticaldimensionsinthebehavioroftheSTD.Wefindthatfordimensionsbelowd1≃26theSTDispurelyconcave(andthustypicalsign-timehisto-riesdominateq(t)),whilefordimensionsaboved2≃46theSTDispurelyconvex(andthusq(t)iscontrolledbyraresign-timehistories).Fordimensionsintherange

1

d1Weconsiderascalarfieldφ(x,t)inad-dimensionalspacewhichistobeevolvedunderthediffusionequation:

∂tφ=D∇2φ,

(1)

wherewehaveφ(x,t=0)=ψ(x).TherandomfieldψistakentohavezeromeanandtobeGaussiandistributedwithcorrelations󰀇ψ(x)ψ(x′)󰀈=∆δd(x−x′).Theresultsthatfollowarecompletelyindependentofthenatureofthedistribution(solongasitdescribesshort-rangecor-relations)andthevaluesofDand∆(solongastheyarepositivedefinite).Wedefinethepersistenceproba-bilityforthissystemastheprobabilitythatthediffusionfieldatagivenpointinspace(theorigin,say)hasneverchangedsignuptotimet.Thesign-timeτ(t)issimi-larlydefinedastheproportionoftimetthefieldattheoriginhasbeenpositive.TheSTDisthehistogramcon-structedfromtheensembleofsign-timescollectedfromalargenumberofrealizations.

Thepersistenceprobabilityq(t)forthediffusionequa-tionhasbeenmeasuredpreviouslyfromexplicitsim-ulationsofadiscretediffusionprocessonafinited-dimensionalgrid[1]fordimensionsd=1,2,and3.Foragivensystemonecollectsthepersistencetimeforeachsiteofthelatticeandformsq(t)bybinningthesevalues.Thereisanimplicitassumptionconcerningself-averagingsincethesepersistencetimesarenotindependent.Thisisequivalenttostatingthatq(t)alsomeasuresthefrac-tionofpersistentsitesinthesystem,whichiscertainlyplausibleinthelargetimelimit.

Wehavemeasuredq(t)andtheSTDusinganalter-nativealgorithmwhichhasmanyadvantagesoverdirectsimulationofthediscretediffusionprocess.Thealgo-rithmisconstructedasfollows.Firstwenotethatthediffusionequationmaybeexplicitlyintegrated.Restrict-ingourattentiontothediffusionfieldattheoriginwehave

󰀁

φ(0,t)=ddyg(y,t)ψ(y),(2)whereg(x,t)=(4πDt)−d/2exp(−x2/4Dt)istheheat

kernel.Thekeyobservationisthatthesolidanglein-

tegrationinEq.(2)maybeabsorbedintoaredefinitionoftherandomfield,asoriginallynotedbyHilhorst[8].DefiningaGaussianrandomfieldΨ(r)withcorrelations󰀇Ψ(r)Ψ(r′)󰀈=δ(r−r′),wehave

󰀁∞φ(0,t)∝

drr(d−1)/2e−r

2

/t

Ψ(r),(3)

0

wherewehavescaledoutDand∆andignoredthe(pos-itivedefinite)prefactor,asonlythesignofφ(0,t)isrel-evanttothepersistentproperties.Tomeasureq(t)fora

givendimensiondweconstructarandomfieldΨ(r)andperformtheradialintegralaboveforaseriesofincre-mentedtimes,untiltheintegralchangessign.WerecordthetimeatwhichthissignchangeoccursandrepeattheprocedureforanotherrealizationofΨ(r).Thiscollectionofpersistencetimesisthenusedtoconstructq(t).IfwewishtomeasuretheSTDweevaluatetheintegralforaseriesofincrementedtimeswithinafixedtimerangeandrecordthefractionoftimeforwhichtheintegralisposi-tive.Thecomputertimerequiredfortheseproceduresisessentiallyindependentofthevalueofd.

Thisalgorithmissuperiortodirectnumericalsimula-tioninmanyrespects:i)anydimensionmaybestudied(evennon-integer),ii)therearenofinitesizeeffects,iii)thereisnoimplicitassumptionconcerningself-averaging,iv)inmeasuringq(t)agivenrealizationisdiscardedassoonasthefieldchangessign(inadirectsimulation,allfieldpointsarestudiedevenifthevastmajorityarenolongerpersistent),v)thealgorithmmaybeoptimizedallowingverylongtimeintervalstobestudiedwithease.Weshallnowdiscussafewtechnicaldetailsconcerningoptimization.Onemaythinkthatinordertomeasureq(t)onemustevaluatetheintegralin(3)onafinegridofincrementedtimes,soasnottomissanyeventinwhichthefieldchangessigntwiceoverashorttimeinterval.Infactthisisnotthecase.ForagivenrealizationofΨwehaveexplicitlyevaluatedthefunctionφ(0,t)onafinemeshoftimeslicesandfinditissmooth,becomingmoresoastimeproceeds.Thisisreasonablewhenoneconsidersthatunderdiffusionthefieldatagivenpointwillhaveincreasinglysmootherdynamicswithtime.Duetotheprogressivesmoothnessofφ(0,t)weneedonlysampleitonalogarithmictimescale.

Thebaseofthelogarithmtobeusedisdeterminedfromcalibration.Wefindtheexactzeroesof104realiza-tionsoverthreedecadesoftimeusingafinelinearmeshoftimeslices.Wethenruntheoptimizedcodewhichsearchesforthesezeroesfirstusingalogarithmicscale,andthenusingrecursivebisectionofalogarithmicinter-valifithasreasontobelieveazerolieswithin.Theloga-rithmbaseischosensothatallofthezeroesfromthe104samplesarefoundcorrectly.Thislimitsthesystematicerrorofourresultstolessthan0.01%.Thecoreofthealgorithmconcernsthedetectionofzeroeswithinaloga-rithmicinterval,basedonthevalueofthefunctionφ(0,t)

attheboundariesoftheinterval.Weusetwotechniques.Thefaster,butlessprecise,algorithmusesrecursivebi-sectionofanintervalonlyifthevaluesofφ(0,t)attheboundarieshavedifferentsigns(inwhichcasetherehastobeanoddnumberofzeroesintheinterval).Themoresophisticatedalgorithmalsousesinformationaboutthederivativesofφ(0,t)attheboundariesoftheinterval.Thisalgorithmwillalsouserecursivebisectiontocheckforazeroifthevaluesofφ(0,t)attheboundarieshavethesamesign,butthederivativeshaveoppositesigns(indicatinganextremumintheintervalandthepossi-bilityofanevennumberofzeroes).Asanexampleofcalibration,ford=1thefirstalgorithmfindsallzeroescorrectlyoverthreetimedecadesifthelogarithmbaseusedis1.1.Thesecondalgorithmfindsallzeroescor-rectlywiththelogarithmbase2.0.(Asdincreasesthesecondalgorithmrequiresasmallerlogarithmicbasetofindallthezeroes–asanexample,ford=20there-quiredbaseis1.2.)Formeasurementsofq(t)wealwaysusethesecondalgorithm,butformeasuringtheSTDwegenerallyusethefirstalgorithm(astheSTDrequiresmuchlongercomputingtime).

Usingthismethodwehavebeenabletomeasureq(t)andtheSTDforawiderangeofdimensions.Fordimen-sionsd=1,2,and3weareabletomakeestimatesforthepersistenceexponentθasshowninTable1,withmuchhigherprecisionthanformerlyobtained[1].InTable1wealsogiveestimatesforθinhigherdimensions,whichuntilnowhavenotbeenmeasured.Bycomparingtheseresultswiththoseobtainedfromtheindependentintervalapproximation(IIA)[1]weseethattheIIAconsistentlyunderestimatestheexponentvalues.

d

θ(d)

0.12050(5)

10.0

2.0

0.6556(2)0.2382(1)

30.0

4.0

0.9312(4)

0.3173(2)50.00.7491(3)

46.0d[1].Byplotting

ourmeasuredexponentsona√log-logscalewehavefittedthisbehaviorandfindθ∼c

thebottomofTable1wehavegivenexponentestimatesforthetwocriticaldimensionsinthissystem.ToclarifytheroleofthesedimensionsweshallnowdiscussourdatafortheSTD’s.

11.00.0060.006f(u)d=50d=40d=30d=200.00550.00550.0050.0050.00450.00450.0040.004q(t)0.10.1d=100.00350.003500.10.20.30.40.50.60.70.80.910.0u1.0FIG.3.STD’sfordimensions10,20,30,40,and50.0.010.0110101001021000100001000001e+06t1e+07103104105106107FIG.1.Persistenceprobabilityq(t)versust,fordimensions(fromtoptobottom)1,2,3,4,and5.

11.0q(t)0.110−10.0110−2Westartbynumericallydeterminingd2.Bysimply

measuringthepersistenceprobabilityq(t)forarangeofdimensionswecanidentifythatdimensionatwhichθ=1.AscanbeseenfromTable1,wefindthatθ(46.0)=1.0010(5).Acloseranalysisleadsustotheresultd2=45.9(1).Thismaybeverified(toalesserprecision)bystudyingtheSTD’s.InFigure4weshowthetailsoftheSTD’sford=45.0andd=46.0.Noteintheformercasetheupturnsignallinga(very)weakintegrablesingularityinthedistribution.0.00360.00360.00355f(u)0.00350.00350.003450.00340.00340.003350.00330.00330.003250.00320.00320.00−30.001100.000110−41e-0510−5d=45.010104105106t1e+071021000103100001000001e+0610100107FIG.2.Persistenceprobabilityq(t)versust,fordimensions(fromtoptobottom)10,20,30,40,and50.

d=46.0TomeasuretheSTD,P(τ,t),weevolvethesystemforthreedecadesoftimeandbylocatingzeroesofφ(0,t)determinethesign-timeτ(t)foraparticularrealization.Thisprocessisrepeatedfor108samplesandtheSTDisconstructedbybinningthesign-times.InRef.[6]itisshownthattheSTDhastheexactscalingformP(τ,t)=(1/t)f(u)whereu=τ/t.InwhatfollowsweshalldescribetheSTDintermsofthescalingfunctionf(u),whichissymmetricaboutu=1/2.ThetailsoftheSTDencodeinformationaboutthepersistenceprob-ability[6].Forexample,thelefttailisexpectedtovaryasuθ−1.Thus,forθ<1theSTDwillhaveintegrablydivergenttails,whereasforθ>1theSTDwillhavevan-ishingtails.Sinceθdivergeswithd,thereexistsacriticaldimensiond2atwhichθ(d2)=1,andwhichseparates(crudelyspeaking)convexandconcaveSTD’s.Fromournumericalworkwefindthatthisscenarioistoosimple,andthatforarangeofdimensionstheSTDhasmorethanoneextremum(asshowninFigure3).

0.020.040.040.060.060.080.080.1u0.1FIG.4.TailsoftheSTDford=45.0andd=46.0.FromtheSTD’sinFigure3weseethereisanother

criticaldimensiond1separatingSTD’swhicharepurelyconcavefromthosewiththreeextrema.Thereisnoob-viouswaytoidentifythisbymeasuringthepersistenceexponent.InsteadwemustusedirectmeasurementsoftheSTD’swhichnaturallyleadstoalesspreciseesti-mate.InFigure5weshowthemeasuredSTD’sforasmallrangeofdimensionsaroundd=26.0.Fromthisdatawearriveattheresultd1=26.0(5).Inanearlierwork[6]d2wasestimatedbyassumingthattheSTDind2dimensionsisperfectlyflat.Thisledtothepredic-tiond2=35.967...,which,withintheerrorbars,isthearithmeticmeanofthemeasuredvaluesofd1andd2.Ourmainresultwithregardtothepersistenceprop-ertiesofthediffusionequationconcernsthenon-trivialbehavioroftheSTDasafunctionofd.FordtailsoftheSTD)isdominatedbytypicalsign-timehis-tories.Ford>d2=45.9(1)theSTDisapurelyconvexfunction,andinthiscasethepersistenceprobabilityisdeterminedfromrareeventsinthesetofsign-timehis-tories.Thereisarangeofdimensionsd10.004760.004760.00474f(u)d=27.00.004720.00472d=26.50.00470.004680.00468d=26.00.00466d=25.50.004640.00464d=25.00.20.30.40.50.60.70.7u0.80.8FIG.5.STD’sforasmallrangeofdimensionsaboutd1≃26.0.Thehorizontallinesareaguidetotheeye.

Clearlyifoneistofindapplicationsofourhigh-dimensionalresults,thenthespaceinwhichthediffusionprocessoccurscannotbetakenasthefamiliarphysicalspace(atleastwithintherealmofclassicalphysics).Averycommonmanifestationofthediffusionequationiswithinthefieldofstochasticprocesses.Indeed,theonedimensionaldiffusionequationisthesimplestpossibleformoftheFokker-Planckequation,describingtheprob-abilitydistributionofasinglerandomwalker[10].ForNindependentrandomwalkersinonedimension,themulti-variateprobabilitydistributionρ(x1,...,xN)sat-isfiestheN-dimensionaldiffusionequation.Givenran-dominitialpositionsofthewalkers,weshouldfocusonthesign-timehistoriesofρwithrespecttoitsmeanvalueovertheensembleofinitialconditions.ThedistributionofthesesigntimeswillhavesharpchangesinbehaviorasthenumberofwalkersNisvariedthroughN1andN2=46.Ifeachwalkerexistsinad

˜=26

dimensionalspace,thentheequationisNd

˜effectivedimensionalityofthediffusion

andthecriticalwalkernumbersN1andN2willbechangedaccordingly.AnalogousstatementsmaybemadeforasystemofNdirectedpolymers(e.g.

magneticfluxlines)inad

˜dimensionalspace,sincethemulti-variatetheNd

˜partitionfunctionforNlinesalsosatisfies

dimensionaldiffusionequation[11].Ourresultsalsoindicatethatpersistentfluctuationsinthephasespaceoffew-bodysystemsmayhavesharptransitionsonvaryingthenumberofdegreesoffree-dom.By“few-body”systems,wehaveinmindorganicmoleculescomposedoftensofatoms,thedynamicalpropertiesofwhicharecurrentlyreceivingagreatdealofattention[12].

InthisLetterwehavepresentedresultsforthepersis-tenceprobabilityandsign-timedistributionforthedif-fusionequationoverawiderangeofdimensions.Theseresultshavebeenobtainedviaanewalgorithmwhichal-lowsextremelyprecisemeasurementsofthepersistenceproperties.Thisalgorithmisbasedontheintegrabilityofthediffusionequation,afeaturesharedbymanyotherin-terestingproblemssuchastheEdwards-Wilkinsonmodelofinterfacegrowth[13],andtheBurgersequation[14]offluidturbulence.Thepersistencepropertiesoftheseandsimilarmodelsmaybeanalyzedtohighprecisionusingthemethodsoutlinedhere.