计算凝聚态物理.ppt

计算凝聚态物理 凝聚态物质的数值模拟方法 5 分子模型 分子动力学方法马红孺 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Molecularsystems Inmostcasestheinteractionpartcanbeapproximatedbypairinteractions OnefamousexampleistheLennard Jonespotential 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Averyimportantquantityinstatisticalmechanicsisthepaircorrelationfunctiong r r0 definedas where Itmayalsobewrittenas 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Forahomogeneoussystemthepaircorrelationfunctiondependsonlyonthedistancebetweenrandr0 Inthiscasewedenoteitasg r Theg r r0 isproportionaltotheprobabilitythatgivenaparticleatpointrandfindanotherparticleatpointr0 Atlargedistanceg r tendsto1 wemaydefinethetotalcorrelationfunction TheFouriertransformoftheabovefunctiongivesthestaticstructurefunction orstructurefactor 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 ThestructurefunctionisdefinedasthecorrelationfunctionofFouriercomponentofdensityfluctuations Thedensityisdefinedas andthedensityfluctuationis anditsFouriercomponentis 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 当体积趋于无限时 红颜色的部分可以略去 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Thestructurefactorcanbemeasureddirectlybyscatteringexperimentsandcanalsobecalculatedbysimulations Manyphysicalquantitiescanbeexpressedintermsofthepaircorrelationfunctions forexampletheenergyinNVTensembleis Thepressureis 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Thecompressibility Thisexpressioncanbederivedfromthefluctuationsofparticlenumbers Sinceso 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Ontheotherhand itcanbeprovedthat Wehavethefinalresult Thetimecorrelationfunctionisthecorrelationsoftwophysicalquantitiesatdifferenttimes Forsystemsatequilibriumthetimecorrelationfunctionisafunctionofthetimedifferenceonlyandcanbewrittenas 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Thevelocityautocorrelationfunctionoftheithparticleis Thiscanbederivedfromthedefinitionrelation wewillbacktothispoint Whichisrelatedtothediffusionconstantoftheparticle whichholdsforlarget 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Ingeneral transportcoefficientisdefinedintermsoftheresponseofasystemtoaperturbation where isthetransportcoefficient andAisaphysicalvariableappearingintheperturbationHamiltonian ThereisalsoanEinsteinrelationassociatedwiththiskindofexpression whichholdsforlarget t where istherelaxationtimeof 2003 10 21 上海交通大学理论物理研究所马红孺 分子模型 Theshearviscosity isgivenby or Here ThenegativeofP isoftencalledstresstensor 2003 10 21 上海交通大学理论物理研究所马红孺 MonteCarlo模拟 MonteCarlosimulationofParticleSystems 粒子系统的MonteCarlo模拟和自旋系统原则上是一样的 Metropolis算法为 1 随机或顺序选取一个粒子 其位置矢量为 对此粒子做移动2 计算前后的能量差 决定是否接受移动 3 在达到平衡后 收集数据 计算物理量 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Moleculardynamicssimulations MDmethodisessentiallytheintegrationoftheequationofmotionoftheclassicalmany particlesysteminaperiodoftime Thetrajectoriesofthesysteminthephasespacearethusobtainedandaveragesofthetrajectoriesgivevariousphysicalproperties SinceweworkonrealdynamicsinMDsimulationswecanalsostudythedynamicpropertiesofthesystemsuchasrelaxationtoequilibrium transportetc 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 ConsiderarectangularvolumeofL1 L2 L3 withNclassicalparticlesputin Theparticlesareinteractwitheachother Inprinciple theinteractionincludepairinteractions threebodyinteractionsaswellasmanybodyinteractions Forsimplicitywewillconsiderhereonlypairinteractions Inthiscaseeachparticlefeelaforcebyallotherparticlesandwefurtherassumetheforceisdependonlyondistancesfromotherparticlesandforeachpairtheforcedirectedalongthelinejointhepairofparticles Sotheforceontheithparticleis whereisanunitvectoralongrj ri 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Periodicboundarycondition PBC whereL arevectorsalongtheedgesoftherectangularsystemvolumeandthesumoverniswithallintegersn Usuallythissumisthemosttimeconsumingpartinasimulation 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 GeneralprocedureofMD NVEensemble 1 Initialize 2 Startsimulationandletthesystemreachequilibrium 3 Continuesimulationandstoreresults 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Initialize 1 Specifythenumberofparticlesandinteraction 2 Setupthesimulationbox 3 Specifythetotalenergyofthesystem 4 Assignpositionandmomentaofeachparticle a InmanycasesweassignparticlesinaFCClattice IfweusecubicunitcellandcubeBOXthenthenumberofparticlesperunitcellis4 andthetotalnumberofparticlesarea4M3 M 1 2 3 ThatiswemaysimulationsystemswithtotalnumberofparticlesN 108 256 500 864 b ThevelocitiesofparticlesaredrawfromaMaxwelldistributionwiththespecifiedtemperature ThisisaccomplishedbydrawingthethreecomponentsofthevelocityfromtheGaussiandistribution 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Thedistributionofthex componentofvelocityis DrawnumbersfromaGaussian Consider Then wherev2 vx2 vy2and 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Sothedistributionofvxandvymaybeobtainedfromvand Thedistributionofv Thedistributionof isuniformintheinterval 0 2 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Generaterandomnumbersforagivendistribution ForagivendistributionP y weconsiderhowtogetarandomnumberydrawfromP y fromarandomnumberxdrawfromuniform 0 1 i e wearegoingtofindafunctionf x fromwhichforaseriesofrandomnumbersxdistributeduniformlyintheinterval 0 1 y f x willdistributedaccordingtoP y 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 then Since Exponentialdistribution 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Thedistributionofv 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Drawrandomnumbers uniformlydistributedintheinterval 0 2 AnothermethodofdrawrandomnumbersintheGaussiandistributionisthroughthefollowingempiricalmethods Considerthedistribution 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Accordingtothecentrallimittheorem ifwedrawuniformrandomnumbersriininterval 0 1 anddefineavariable whenn 1thedistributionof istheGaussiandistribution Ifwetaken 12 weget 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Afterthegenerationofthevelocityofeachparticle wemayshiftthevelocitysothatthetotalmomentumiszero ThestandardVerletalgorithmisthefirstsuccessfulmethodinhistoryandstillwideusedtodayindifferentforms Itis Tostarttheintegrationweneedr h givenby 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Variationsofthismethodare And Bothofthesevariationsaremathematicallyequivalenttotheoriginalonebutmorestableunderfiniteprecisionarithmetic 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Thetemperatureofthesystemisgivenbytheequalpartitiontheorem thatistheaverageofkineticenergyofeachdegreeoffreedomishalfkBT TheN 1isduetotheconservationofthetotalmomentumwhichreducethedegreeoffreedomby3 Toreachthedesiredtemperaturewemayscalethevelocityateveryfewstepsofintegration 2003 10 21 上海交通大学理论物理研究所马红孺 分子动力学模拟 Afterthesystemreachtoequilibriumtheintegrationcontinueinthesamemethodasabovewithoutscalingofvelocity Thedataarestoredoraccumulatedforthecalculatingphysicalproperties ThestaticpropertiesofphysicalquantityAisgivenbytimeaverage 200