文档介绍：该【finite-barrier corrections for multidimensional barriers in colored noise thomas bartsch资料 】是由【赖大文档】上传分享，文档一共【14】页，该文档可以免费在线阅读，需要了解更多关于【finite-barrier corrections for multidimensional barriers in colored noise thomas bartsch资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..PHYSICALREVIEWE99,052211(2019)Finite-barriercorrectionsformultidimensionalbarriersincolorednoiseThomasBartsch,1,*,2,3,?,2,?,4,§1CentreforNonlinearMathematicsandApplications,DepartmentofMathematicalSciences,LoughboroughUniversity,LoughboroughLE113TU,England,UnitedKingdomplejos,íaAgronómica,AlimentariaydeBiosistemas,icadeMadrid,AvenidaPuertadeHierro2-4,28040Madrid,Spain3InstitutodeCienciasMatemáticas,Cantoblanco,28049Madrid,Spain4DepartamentodeQuímica,UniversidadAutónomadeMadrid,Cantoblanco,28049Madrid,Spain(Received4August2018;published15May2019)Theusualidenti?cationofreactivetrajectoriesforthecalculationofreactionratesrequiresverytime-consumingsimulations,,,usingthisperturbativescheme,:.(DOF?s)andforheatbathsthataremodeledbywhiteTransitionstatetheory(TST)essful(memoryless)aswellascolored(correlated)noise,.,--[1?4]TSThasprovidedapowerfulconceptualframeworkforplicationtotherealisticisomerizingreactionLiNC-dimensionalofactivatedprocesses,thatproceedfromsuitablyde?nedapproachcanbefoundinRef.[20].?reactant?to?product?states[5?14](LE)[21?23].(DS)betweenreactantandproductregionsofphasetunneling,whichcanbeimportantinthecaseoflightparticles[24],andtheinteractionwithelectronicexcitedstatesthroughspaceischosenclosetothebarriertop,thereactionratecanputedfromthesteady-state?uxoftrajectoriesthroughconicalintersections[25].Moreimportantlyforourpurposes,,TSTmakestwosimplifyingassump-tions:?rst,thateverytrajectorythatcrossestheDSfromthecorrelatedwithitselforwiththereactivesystem,andthatthereactanttotheproductsideisreactiveor,equivalently,,and,second,,andthiscorrelationwilldecayonIfeitheroftheseassumptionsisviolated,suitablecorrec-,theno-recrossingassumptionposes(GLE)(seeRefs.[21?23,26]).particularchallengesbecause,unlessaDSissoughtinanThephasespaceoftheGLEhasin?-in?nite-dimensionalextendedphasespace[15],therandomlyever,ifthefrictionkernelsatis?esalineardifferentialequa-?uctuatingforceexertedbythebathwillcausethesystemtotionwithconstantcoef?cients,theGLEcanberephrasedcrossanychosenDSpotentiallymanytimes,inparticularinasastochasticdifferentialequationona?nite-,arecrossing-freeDSphasespace[27].Themethodsproposedinthispaperapplytocannotbefoundinthermalsystemswithanharmonicbarrierskernelsofthistype,thoughforsimplicityonlyanexponential[16](asphasereactionsathighenergies[17?19]).frictionkernelwillbediscusseddirectly[seeEq.(12)below].Itisthischallengethatweproposetoaddressinthispaper,,anyfrictionkernelthattendstozeroforlargetimescanbe*T.******@?fabio.******@-Weierstrasstheorem[28].?rosamaria.******@§f.******@-0045/2019/99(5)/052211(14)052211-1?2019AmericanPhysicalSociety:..BARTSCH,REVUELTA,BENITO,ANDBORONDOPHYSICALREVIEWE99,052211(2019)-.[29]formanifoldalsoexistsinareactivesystemdescribedbyaGLEsystemswithwhitenoise,andcorrectionstothetransmissionwithGaussiancorrelatednoise,andwewilluseittoderivefactorwereobtainedbyPollakandTalknerinRefs.[30?32],evenwhentheyhavemanyDOF?-dimensionaltoturnovertheory[33?36].Differentmethodshavebeenbarrierandapotential[42,43,45,46]withtwoDOF?,plexityofsystemsofhigherdi-sometheorieswerebasedontheRayleighmethod[29],[30?32]-?nitebathofharmonicoscillators[37].Additionally,-,Ref.[putesthereactionratesofaone-??ux,-,,inpractice,,tothebestofourknowledgeareactionthatInthissection,wesummarizethefundamentalsofreactionshowstheKramersturnoverinarealisticsettingwas?.[39].detaileddiscussion,see,forexample,Refs.[21?23].Moreover,weavoidusinganexplicitmodeloftheheatLetusconsiderasystemwithdDOF?sdescribedbythebaththatintroducesanin?nite--d-ponentsofwhichstead,weworkdirectlyinthephasespaceoftheGLE,whichwillbedenotedbyxn,withn=0,1,...,d?1,wheren=0is?=1,...,d?=xDS,putationalandconceptual0pointsofview,ovisualizetherelevantandthereactantandproductregionsarede?nedbyx0<>x0,respectively,thereactionrateisgivenbytheofthispaperisadetaileddescriptionofthegeometrical?ux-over-=,(1)Nofpapers[20,40?46]?=?1phasespacecoordinates:thevelocityinthevicinityofthebarriertopforalltimes,withouteverv0perpendiculartothesurfaceandthecoordinatesx⊥=(x1,x2,...,xd?1),andvelocitiesv⊥=(v1,v2,...,vd?1)?uxthenequalsthecaseofaharmonicbarrier,esnoiselessJ=vχ(v,x,v),(2)ifthedynamicsisstudiedinatime-dependentcoordinate0α0⊥⊥α,-noiseandoverastationary-(IC?s)ontheDS.[Inthede?nitionofEq.(2)?uxwehaveomittedirrelevantconstantfactorsthatwillasymptoticallyapproachtheTStrajectoryforlongtimesanditcancelinthetransmissionfactorofEq.(5).]Thecharacteristicseparatestrajectoriesthatdescendintotheproductwellinthefunctionχαtakesthevalue1ifthetrajectorygivenbytheICdistantfuturefromthosethatdescendintothereactantwell.(xDS,v0,x⊥,v⊥)?uxifitactuallyleadstoareaction,.,-.[45,46]-2:..FINITE-BARRIERCORRECTIONSFOR?PHYSICALREVIEWE99,052211(2019)StandardTSTsidestepsthedynamicalproblembyassum-Withnolossofgenerality,=-words,TSTassumesthecharacteristicfunctionχTST(v0)=neckforreactivityisformed,andintheregimeofmoderate(v0),??cally,therelevantneighborhoodJ=v0χ(v0)v0(3)ofthesaddlepoint,whichinourapproachisrepresentedbyleadstoanapproximationtothereactionrate,theareaexploredbytheTStrajectory,hasanextensionof√TSTorderkBT,,ifthepotentialTSTJisexpandedasaTaylorseriesaroundthesaddlepoint,eachk=,(4)Nordercanberegardedasbeingsmallerthanthepreceding√-recrossingassumptionismea-expansionparameterεtore?-keepingdevice;the?nalrateformulaswilldependonkthecoef?=TST<1.(5)AfundamentalassumptionofratetheoryisthatthethermalparedtothebarrierheightUnlessthefrictioncausedbytheheatbathisveryweak,V?,measuredfromthebottomofthereactantwelltothethestationary-statedistributionofIC??,aswasdone,foralwaysbemadeintheratecalculationspresentedhere,thoughexample,inRefs.[29?31].However,thebarrierheightisathedynamicaltheoryattheheartofthispaperdoesnotrequirenonlocalfeatureofthepotential,plicatedpoten-itbecausetheexactexpressionforthecharacteristicfunctiontialthevariousexpansioncoef??sisthenperformedoveranensemblewithmoderatetostrongfrictionthereactionrateisdeterminedbyprobabilitydensityspatialdiffusionintheneighborhoodofthebarrier,andthe2Taylorcoef?cientsdirectlycharacterizethisneig