Computation of free energy profiles with parallel

arXiv:cond-mat/0611276v3[cond-mat.stat-mech]16Jan2007APS/ComputationoffreeenergyprofileswithparalleladaptivedynamicsTonyLeli`evre,∗MathiasRousset,†andGabrielStoltz‡CERMICS,EcoleNationaledesPontsetChauss´ees(ParisTech),6&8AvenueBlaisePascal,77455Marne-la-Vall´ee,France.(Dated:February4,2008)AbstractWeproposeaformulationofadaptivecomputationoffreeenergydifferences,intheABFornonequilibriummetadynamicsspirit,usingconditionaldistributionsofsamplesofconfigurationswhichevolveintime.Thisallowstopresentatrulyunifyingframeworkforthesemethods,andtoproveconvergenceresultsforcertainclassesofalgorithms.Fromanumericalviewpoint,aparallelimplementationofthesemethodsisverynatural,thereplicasinteractingthroughthereconstructedfreeenergy.Weshowhowtoimprovethisparallelimplementationbyresortingtosomeselectionmechanismonthereplicas.Thisisillustratedbycomputationsonamodelsystemofconformationalchanges.PACSnumbers:05.70.Ln,02.70.Ns,02.50.Ey1I.INTRODUCTIONOneofthemostimportantgoalsofmolecularsimulationisthecomputationoffreeenergydifferencesasafunctionofsomeselecteddegreesoffreedom,calledreactioncoordinates.Thedynamicsofthesystemcanindeedoftenbesplitintoslowlyevolvingdegreesoffreedom,whichdeterminethereactioncoordinatestobeused,andotherrapidlyevolvingdegreesoffreedom.Thefreeenergydifferencesallowtocharacterizeglobalchangesinthesystemunderstudy,andgiveinformationabouttherelativestabilitiesofseveralspecies,aswellastheirtransitionkinetics.However,thefreeenergybarrierstoovercomearesolargeinmanyapplicationsthatacomputationbasedonastraightforwardsamplingisunfeasiblesincethesystemremainsstuckinmetastablefreeenergysets.Aclassicaltechniquetocomputefreeenergydifferencesisthermodynamicintegration,datingbacktoKirkwood1,whichmimicsthequasi-staticevolutionofasystemasasucces-sionofequilibriumsamplings,whichamountstoaninfinitelyslowswitchingbetweentheinitialandfinalstates.Anotherclassicaltechniqueisthefreeenergyperturbationmethod,introducedbyZwanzig2,whichrecastsfreeenergydifferencesasaphase-spaceintegral,sothatusualsamplingtechniquescanbeemployed.Noticealsothatthereexistmanyrefine-mentsforthosetwoclassesoftechniques,suchasumbrellasampling3.Morerecently,methodsrelyingonnonequilibriumdynamicshaveemerged.TheyfollowthepioneeringworkofJarzynski4,orusesomeadaptivedynamicssuchastheWang-Landauapproach5,theadaptivebiasingforce(ABF)6,7,8,orthenonequilibriummetadynamics9.Theseapproachesusethewholehistoryoftheexplorationprocesstobiasthecurrentdy-namicsinordertoforcetheescapefrommetastablesets.Thisisdonebysimultaneouslyestimatingthefreeenergyfromanevolvingensembleofconfigurationsofthedynamics,andusingthisestimatetobiasthedynamics,sothattheeffectivefreeenergysurfaceexploredisflattened.Inthelongtimelimit,thebiasexactlygivestheactualfreeenergyprofile.Adap-tivemethodscouldthereforebeseenasumbrellasamplingwithanevolvingpotential.Thiswasalreadynoticedinapreviousstudypresentinganadaptivedynamicsasa’self-healingumbrellasampling’10.Topresenttheadaptivemethodsmentionedaboveinageneralandunifyingframework,itisconvenient,asisdonein9,toconsiderensembleofrealizations(seeEq.(4)).Thesystemisthendescribedbythedistributionoftheconfigurationsofthisensembleinthe2limitofaninfinitenumberofreplicassimulatedinparallel.Thekeypointistoreformulatethecomputationofthebiasofadaptivedynamics,usingconditionaldistributions(thatis,distributionoftheconfigurationsforagivenvalueofthereactioncoordinate)ofthelattersample.Thiswasalreadyproposedin11intheequilibriumcase,andissomewhatimplicitin9.Thisconceptclarifiesthepresentationofadaptivemethods,allowsmathematicalproofsofconvergence12oratleast,existenceofastationarystateofthedynamics(stillinthecaseofaninfinitenumberofreplicas),andsuggestsnaturalnumericalstrategies:thediscretizationmaybedonethroughaparallelimplementationofseveralreplicasofthesystem,whichallcontributetoconstructthefreeenergyprofile.Suchaparallelimplementationwasalreadyproposedin13inthecaseofmetadynamics.Weshowherehowanadditionalselectionprocessonthereplicascanenhancethesamplingofthereactioncoordinatesincomparisonwithastraightforwardparallelimplementation.Thepaperisorganizedasfollows.InSectionII,wedescribethegeneralformalismforadaptivedynamics,usingconditionalprobabilities,andshowhowtoupdatethebiasingpotentialinordertocomputethefreeenergyprofileinthelongtimelimit,usingafixed-pointstrategy.SomeapplicationsofthisformalismarepresentedinSectionIII,andallowtorecovertheusualadaptivedynamicssuchasthenonequilibriummetadynamics,theWang-LandauschemeortheABFmethod.WethendiscusspossibleparallelimplementationstrategiesinSectionIV.Inparticular,itisshowninSectionIVBhowaselectionprocesscanenhancethestraightforwardparallelimplementation.ThisisfinallyillustratedbynumericalresultsforatoymodelofconformationalchangesinSectionV.II.AGENERALFRAMEWORKFORADAPTIVEDYNAMICSA.NotationsForasystemdescribedbyapotentialV(q),theBoltzmannmeasureinthecanonicalen-sembleisZ−1exp(−βV(q))dq(whereZisanormalizationconstant,theso-calledpartitionfunction).Considerareactioncoordinateξ,takingvaluesintheonedimensionaltorus,orintheinterval[0,1].Inthelattercase,