Random walks and the Metropolis Algorithm

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1、Chapter9RandomwalksandtheMetropolisalgorithmNelmezzodelcammindinostravita,miritrovaiperunaselvaoscura,chéladirittaviaerasmarrita.(DivinaCommedia,Inferno,CantoI,1-3)DanteAlighieriThewaythatcanbespokenofisnottheconstantway.(TaoTeChing,BookI,I.1)LaoTzu9.1Motivatio

2、nInthepreviouschapterwediscussedtechnicalaspectsofMonteCarlointegrationsuchasalgorithmsforgeneratingrandomnumbersandintegrationofmultidimensionalintegrals.Thelattertopicservedtoil-lustratetwokeytopicsinMonteCarlosimulations,namelyaproperselectionofvariablesandi

3、mportancesampling.Anintelligentselectionofvariables,goodsamplingtechniquesandguidingfunctionscanbecrucialfortheoutcomeofourMonteCarlosimulations.Examplesofthiswillbedemonstratedinthechaptersonstatisticalandquantumphysicsapplications.Herewemakeadetourfromthismai

4、nareaofapplications.Thefocusisondiffusionandrandomwalks.TherationaleforthisisthatthetrickypartofanactualMonteCarlosimulationresidesintheappropriateselectionofrandomstates,andtherebynum-bers,accordingtotheprobabilitydistribution(PDF)athand.Withappropriatethereis

5、howevermuchmoretothepicturethanmeetstheeye.SupposeourPDFisgivenbythewell-knownnormaldistribution.Thinkofforexamplethevelocitydistributionofanidealgasinacontainer.Inoursimulationswecouldthenacceptorrejectnewmoveswithaprobabilityproportionaltothenormaldistributio

6、n.Thiswouldparallelourexampleonthesixthdimensionalintegralinthepreviouschapter.However,inthiscasewewouldenduprejectingbasicallyallmovessincetheprobabilitiesareexponentiallysmallinmostcases.Theresultwouldbethatwebarelymovedfromtheinitialposition.Ourstatisticalav

7、erageswouldthenbesignificantlybiasedandmostlikelynotveryreliable.Instead,allMonteCarloschemesusedarebasedonMarkovprocessesinordertogeneratenewrandomstates.AMarkovprocessisarandomwalkwithaselectedprobabilityformakingamove.Thenewmoveisindependentoftheprevioushisto

8、ryofthesystem.TheMarkovprocessisusedrepeatedlyinMonteCarlosimulationsinordertogeneratenewrandomstates.ThereasonforchoosingaMarkovprocessisthatwhenitisrunforalongenoughtimest