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연구자들은 기존의 몬테카를로 메서드 (HMC)가 비트맵 메모리 디스크에 저장되는 반면, 새롭게 개발된 하이 브릿지 메모리 디스크 (HMD)를 사용하여 lattice gauge theory를 해결했습니다. HMC는 1-step size의 계산을 수행하여 속도 향상을 얻을 수 있지만, HMD는 2-3 step size의 계산을 수행하여 속도 향상을 얻습니다.
연구자들은 HMD 알고리즘을 이용하여 Wilson fermion lattice gauge theory를 해결하였으며, 결과가 HMC와 일치하는지 확인했습니다. 또한, 연구자들은 HMD 알고리즘에 발생할 수 있는 오류 (H_errors)가 시스템의 물리적량에 미치는 영향을 분석하였습니다.
결과적으로, 연구자들은 HMD 알고리즘은 HMC 알고리즘보다 더 빠른 속도와 더 정확한 결과를 제공하는 것을 확인하였습니다. 또한, 연구자들은 H_errors가 시스템의 물리적량에 미치는 영향을 분석하였으며, H_errors가 더 이상 시스템의 물리적량을 바꾸지 않는다는 결론을 얻었습니다.
연구결과는 HMD 알고리즘의 정확성과 효율성을 보여주며, lattice gauge theory 연구자들에게 새로운 도구와 방법을 제공할 것으로 기대됩니다.
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OntheDynamicsofLightQuarksinQCDKhalilBitar,RobertG.Edwards,UrsM.Heller,andA.D.KennedySupercomputerComputationsResearchInstituteFloridaStateUniversityTallahassee,FL0-0,USAInternet:fkmb,edwards,heller,adkg@scri.fsu.eduWedescriberecentresultsconcerningthebehavioroflatticeQCDwithlightdynamicalWilsonandStaggeredquarks.Weshowthatitispossibletoreachregionsofparameterspacewithlightpionsm0:=ausingWilsonfermions.IftheHybridMolecularDynamics(HMD)algorithmisusedwiththesameparametersitgivesincorrectresults.Wealsopresentpreliminaryresultsusingahigher-orderintegrationscheme..IntroductionInthispaper,wesetoutwiththeobjectiveofexaminingtheeectsofHerrorsintheintegra-tionofHamilton'sequationsofmotion.WehaveperformedtestsusingavorsofWilsonfermionsat=:0,=0:and0:onalattice.WehavealsostudiedfouravorsofStag-geredfermionsat=:0,mq=0:00onanlattice.Previousstudiesofstep-sizeer-rorsincludingfermionswerelimitedtorelativelysmalllatticesandlargepionmasses.Ourgoalistoextendthesestudiestoparameterregionsthatwillberealisticonfuturemachines.ThestandardmethodsofsimulatingQCDwithdynamicalfermionsareHybridMonteCarlo(HMC)viathe-algorithm[]andHybridMolec-ularDynamics(HMD)viatheR-algorithm[].Themajordistinctionbetweenthetwoalgo-rithmsishowthefermionicdeterminantissimu-latedandhowthecongurationobtainedbyap-proximatelyintegratingHamilton'sequationsofmotionishandled.Thenomenclatureissome-whatconfusing.TheexpressionHybridMolec-ularDynamicsisprincipallyusedtonametheR-algorithm;however,itisalsousedtodescribethe-algorithmwithouttheacceptancetest.Theprincipleadvantageofthe-algorithmwiththeacceptancetest(HMC)isthatthealgorithmisexact{nonitestep-sizeerrorsareintroducedintotheequilibriumprobabilitydistribution.OurtestsarelimitedtotheHybridMonteCarloalgorithmwherewecancomparetheal-SpeakerattheconferencegorithmwithandwithouttheMetropolisaccep-tance.WewillcallthealgorithmwithouttheMetropolisacceptancetheHMDalgorithm.AllourcomputationsweredoneontheConnectionMachineCM-atSCRIusingaconjugategra-dientlinearequationsolver(writteninCMIS)whichrunsat>Gigaops.AmajordicultyinsimulatingQCDisthelongexponentialautocorrelationtimesforthesystemtodecayintoequilibriumandthelongintegratedautocorrelationtimeswhileinequilib-rium.Infact,ourresultsforWilsonfermionswiththelightestpion(m0:=a)indicateadecaytimeofabout00unit-lengthtrajectories.Weshouldexpectthisbehavior.Thepionisthelight-estexcitationofQCD,becauseitistryingtobeaGoldstoneboson.Unlikethequenchedapprox-imationwherethelightestdynamicalexcitationisaglueball,forfullQCDthepionmassistheinversecorrelationlength,m==.Thus,form=0:=aweexpectthatthecorrelationlengthainequilibrium;itisonlyreasonabletoex-pectanalgorithmtotakemuchlongertodevelopsuchlongdistancecorrelationsthanittakestoequilibrateaquenchedsystemwitha.The-oretically,weexpectthecharacteristicrelaxationtimetobethesameastheautocorrelationtime,whichinturnisproportionaltozwherezisthedynamicalcriticalexponent.ForHMCitisbe-lievedthatz[].Howcanwemeasurethecorrelationlengthofanensembleofcongurations?Theobviousmethodofmeasuringistomeasurethepionmassbymeasuringtheasymptoticdecayofapion
correlationfunctionh(0)(t)i.Ifwehaveasetofequilibratedcongurationsthisisacompletelyvalidprocedure,butconsiderwhatwouldhappeniftheMarkovprocesshadnotyetconvergedtothetrueequilibriumdistribution.Becausethesystemstartsfromacongurationwhichdoesnothavethetruelongdistancecorrelationsbuiltin,theunequilibratedcongurationswilltendtohave<=m.Roughlyspeakingwecansaythattheycorrespondmorecloselytotheequilib-riumdistributionofQCDwithalargervalueofthequarkmassmq{weshallcallthisthe\sea"quarkmass{thantheactualvalueappearingintheaction.Ofcourse,theactualdistributionofsomesetofunequilibratedcongurationsissomecomplicatedmess,butitiscertainlyreasonabletoassumethatsuchashiftinmqisthedominanteectwhennearmcrit=0forStaggeredfermionsorcinthecaseofWilsonfermions.Thequarkmassdoesnotonlyappearintheaction,itisalsoexplicitlypresentinthepionop-eratorusedtomeasurethepionmass.Sincethisquarkmasshasnothingtodowiththedy-namicsofthesystemweshallcallitthe\valence"quarkmass.Forourcomputationsweexpectthattheseamqapproachesthevalencemqfromaboveasthesystemapproachesequilibrium:inotherwordsthecorrelationlengthestimatedusingthecorrelationfunctionfora\valence"pionwillbelargerthantheactualcorrelationlengthofthesystemuntilthesystemreachesequilibrium.Foraquenchedcomputationorforaninexactdynamicalquarkalgorithmvalenceandseapa-rametersaredierenteveninequilibrium.Oneillustrationoftheseideasisthatitiseasytomea-surealightpionmassonaquenchedcongura-tion,eventhoughthecorrelationlengthismuchsmallerthantheinversepionmass.Theexistenceofalightvalencepionisnotev-idenceforasystemcontaininglightdynamicalpionsunlessthesystemcanbeshowntohaveequilibratedtothecorrectprobabilitydistribu-tion.Agoodindicatorwehavefoundisblockedgaugeeldoperators(usingTeperblocking),forexampletheblockedspatialplaquette.Blockedplaquettesdonotneedanextraparameterandaresensitivetothelongerrangepropertiesofthesystem.TheeectsofdynamicalquarksshouldappearasamodicationofthequenchedQCDbetafunctionatlongerdistances.Theblockedobservablesindicatecoherenceandcorrelationinthesystem.Theproceedinggeneralargumentsareapplica-bletobothWilsonandStaggeredfermions.Thedierencebetweenthetwofermionsisatwhichquarkmassscaleshouldthestep-sizeerrorssig-nicantlyeecttheequilibriumprobabilitydistri-bution.Thefermioniccontributiontothemolec-ulardynamicsforceisMyMUMyMMyM()whereM=M(U(t)),U(t)isthegaugeeldcongurationatintegrationtimetandisthepseudo-fermioneld.Whenthequarkmassap-proachescriticalitythefermionicmatrixbecomessingularandthesmallesteigenvaluedominatestheforce.Thelowesteigenvalue0ofMyMisrelatedtothepionmass,hencewearguethatthestep-sizeerrorsintheintegrationareampliedbysomefunctionof=m.Wilsonfermionsdonothavechiralsymmetryhence0canwanderarbi-trarilyclosetozeroduringintegration.Ontheotherhand,forStaggeredfermions0>0forallcongurationswithmq>0.Mostlikelythefunc-tionaldependenceon=misdierentforWilsonandStaggeredfermions,butinanycaseweex-pectverysmallpionmassesandasmallseaquarkmasswillberequiredforStaggeredfermionsbe-forethemagniedstep-sizeerrorswillbeanissue.ForourWilsonfermiontests,therstexercisewehadtoundertakewastondasuitablevaluefor.Weranat=0:0,0:,0:,and0:0:ourresultsindicatethatm<0:=aforthersttwovalues,indicatingthatcisprob-ablyinthisregion.Ifweextrapolaterecentlypublishedresults[]wendthatc0:,inroughagreementwithourdata.For=0:0,0:and0:wendthatm0:=a,0:=aand0:=a,respectively.InFigureweshowthepionmassandac-ceptancerateasafunctionofMDtimeforourrunat=0:.Here,RMDistheaccuracyofthenormalizedresidualvectorduringintegra-tion.Itisclearthesystemhasalargeexponen-tialautocorrelationtimeandprobablyhasonly
Figure.Timehistoryoftheeectivepionmass(distance)for=:0,=0:and=0:00onalattice.Symbolsare:()HMCatRMD=0beforetrajectory0andRMD=0after0,(+)HMDatRMD=0and()HMDatRMD=0.Alsoshownistheblocksize=0()HMCacceptanceuncorrectedforautocorrelations.justreachedequilibrium.ItisalsoclearthatasthecorrelationlengthgrowstheHMCacceptanceratefalls,whichrequiresreducingtheintegrationstep-sizeand/ortheconjugategradientaccuracyinordertokeepareasonableacceptancerate.Apreliminarystudyofthesystemindicatesthatthisisprimarilyduetotheincreasingeectofhighfrequencyuctuationsinthefermioniccon-tributionstotheaction.InallourWilsonruns,weusedalinearextrapolationoftheproceedingtwosolutionsfortheinitialguessoftheconjugategradientinverter.Wedecideditwouldbeusefultoseewhatre-sultswewouldgetifweusedtheHMDalgo-rithminsteadofHMCforthissystem(i.e.,ifweomittedtheMonteCarloaccept/rejectstepforRMD=0).Itiscommonlyacceptedthatthisshouldintroducesmallerrorsinphys-icalquantitiesoforder;tooursurprisetheHMDalgorithmwiththesameparametersgavewronganswersforthepionmass.Wealsoper-formedaHMDtestusingRMD=0andfoundthealgorithmgavecompletelywronganswersforthepionmass.Infactotherhintsthatthesys-temcontainedlightpions,suchasthelargenum-berofCGiterationsrequiredpersteptoreachagivenresidual,alsorapidlydisappearedwhenrunningtheHMDalgorithmatRMD=0.Noshownisatestofwhenwereducedthestepsizeto=0:00atRMD=0,theHMDresultswerevirtuallyidenticalto=0:00indicatingtheHerrorsweredominatedbytheCGaccu-racy.Wehaven'tperformedanyHMDtestsatRMD=0;however,weexpecttheCGac-curacytomuddytheresults.Anevenclearerin-dicatorofHerrorsintroducedisviatheblockedspatialplaquette.At=0:and0:andusingRMD=0,theblockedplaquetteforHMDdeviatedsystematicallyfromHMCwhentheacceptancetestwasturnedowhilethepionmassfor=0:withHMDdeviatedonlyslightlyfromHMC.Thesetestsshowthat,CGaccuracyandtheacceptanceareallverycrucialforthealgorithmtogivecorrectphysics.Testsofonlystep-sizeextrapolationsarenotcompleteenoughsincetheCGaccuracyisimportantfornotjustthemini-mizationofHerrorsbutforreversibilityaswell.TheparametersusedinourStaggeredsimula-tionswerebasedontheworkoftheMTccollabo-ration[].Weusedanlatticeat=:0andmq=0:00andobservedm=0:()andm:0.ForHMC,weusedRMD=0andalinearextrapolationoftheinitialguess.ToemphasizetheHerrorsweusedRMD=0forHMDwithazeroinitialguesstomaintainreversibility.Wedidn'tndanystatisticallysig-nicanteectsbetweenHMDandHMCafter00and000trajectorieseach,respectively.Thepionmasshereistoosmallandtherhoistoolargetoberealisticofthecontinuumlimit,however.Withthissmalllattice,weprobablyshouldn'tex-pecttheseaquarkmasstobeverysmall.WearecurrentlyrunningattheMTcparametersof=:andmq0:0onalatticewhichwehopewillprovideatestmoreusefulforcalculationsonfuturemachines.Inaseparatestudy,wecarriedoutamoreaccu-
rateMDintegrationwithWilsonfermionsusingthehigher-orderintegrationschemes[],withthehopeofndingacheapwayofincreasingtheac-ceptancerateforHMC.TheoreticalanalysesofthesealgorithmswerebasedupontheideathatbecausethechangeinenergyoveratrajectorywouldbeH=O()insteadofH=O()onecoulduseamuchlargerstepsizetogetthesameacceptancerate.Thisdoesnotworkinpractice:whiletheCampostrinischemedoesin-deedgivemuchsmallervaluesforH,theinte-grationbecomesunstableifanyattemptismadetoincrease.Indeed,theCampostrinimethodprovedunstableunlessthestepsizewasreducedslightlyfromthatusedinourHMCruns.Thisis,ofcourse,thebehaviorweshouldhaveex-pected.Thelimitonthestepsizeisgivenbytheconstraintfromstability(!where!isthehighestfrequencyofthephysicalsystem)andnotbytheerfc(constpV)dependenceoftheacceptancerateonthevolume(atleastforthevolumeswehaveused).Initiallywethoughttheextracomputationre-quiredfortheCampostrinimethodwassmallbe-causetheinterpolatedsolutionsofthepreviousCGsolutionswereveryaccurate.Thiswaserro-neous,aswehadoverlookedthefactthatthedel-icatecancellationsrequiredbytheCampostrinimethodrequiredamuchmoreaccuratemeasure-mentofHforeachstep,andthusamuchlowerCGresidual.WhenweloweredtheCGresidualtothevaluerequiredtomaketheCGsolutionerrorcontributiontoHsmallerthanthetruestep-sizeerrors,wefoundthatthemagnitudeofHwasgreatlyreducedbutatthecostofmanymoreCGiterations.Sadlywefoundthatunlessenoughex-traCGiterationswereperformedtheuseofinter-polatedinitialguessesfortheCGsolverinducedanirreversibilityintheHMCalgorithmwhichledtoclearlyincorrectresults.OurconclusionsonthisissuearethatonemusteithermakesurethattheCGresidualissmallenoughtoavoidsuchsystematicbiasesoronemustuseatime-symmetricinitialguessfortheCGsolutionsuchaszero.BothoptionsmaketheCampostrinimethodmuchmoreexpensivethansimpleleapfrogintegration.Whichmethodisul-timatelycheaperisstillunderinvestigation.IfwehadcarriedoutanHMDcalculationwithsimilarHerrorsusedinthepresentHMCcom-putations,wewouldhavebeenforcedtousealargervalueofinordertomeasurealightpionmass.Thismasswouldhavebeenmuchsmallerthantheinversecorrelationlengthactu-allypresentinthecongurations.Indeed,wemaywellhavebeenmeasuringvalenceobserv-ablesonanalmost-quenchedsystem,sowewouldhaveconcludedthatphysicswithlightdynamicalquarkswasverysimilartoquenchedphysicswiththesamemasses.Wewouldalsohavefoundamuchsmallerrelaxationtimeandautocorrelationtime,andthusbeenmisledintothinkingthatfullQCDcalculationweremuchcheaperthantheyreallyare.WhataretheimplicationsofthisforstaggeredquarkscalculationsdoneusingtheHMDalgo-rithm?WhileourWilsonquarkresultsdonottellusdirectlyaboutthebehaviorofstaggeredquarks|theeectsweseemightpossiblyjustbeartifactsofWilsonquarkdynamics|theyleadustosuggestthatitisnecessaryforacare-fulzerostep-sizeextrapolationtobedoneforanyHMDcalculation,withspecialattentionrequiredtoverifythatthesystemistrulyinequilibrium.REFERENCESA.D.Kennedy,Intl.J.Mod.Phys.C,();J.C.SextonandD.H.Weingarten,Nucl.Phys.B0().S.Gottlieb,W.Liu,D.Toussaint,R.L.Renken,R.L.Sugar,Phys.Rev.D().R.Gupta,C.Baillie,R.Brickner,G.Kilcup,A.PatelandS.Sharpe,Phys.Rev.D().E.Laermann,R.Altmeyer,K.D.Born,W.Ibes,R.Sommer,T.F.WalshandP.M.Zer-was,Nucl.Phys.B(Proc.Suppl.)(0),Lattice.A.D.KennedyandB.Pendelton,Nuc.Phys.B0(Proc.Suppl. )();S.Gupta,Phys.Let.B,();Intl.J.Mod.Phys.C,().M.CampostriniandP.Rossi,Nuc.Phys.B,(0);M.CreutzandA.Gocksch,Phys.Rev.Let.,().
출처: arXiv:9212.034 • 원문 보기