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MolecularKineticsinCondensedPhases

Theory,Simulation,andAnalysis

RonElber

W.A."Tex"MoncriefJr.EndowedChair OdenInstituteforComputationalEngineeringandSciences andDepartmentofChemistry

TheUniversityofTexasatAustin USA

DmitriiE.Makarov

TheUniversityofTexasatAustin USA

HenriOrland InstitutdePhysiqueThéorique CEASaclay France

Thiseditionfirstpublished2020

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Names:Elber,Ron,author.|Makarov,DmitriiE.,author.|Orland,Henri, author.

Title:Molecularkineticsincondensedphases:theory,simulation,and analysis/RonElber,DmitriiE.Makarov,HenriOrland.

Description:Firstedition.|Hoboken,NJ:Wiley,[2020]|Includes bibliographicalreferencesandindex.

Identifiers:LCCN2019024971(print)|LCCN2019024972(ebook)|ISBN 9781119176770(hardback)|ISBN9781119176787(adobepdf)|ISBN 9781119176794(epub)

Subjects:LCSH:Chemicalkinetics–Mathematicalmodels.|Stochastic processes–Mathematicalmodels.|Molecularstructure.|Molecular theory.

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Contents

Acknowledgments xiii

Introduction:HistoricalBackgroundandRecentDevelopmentsthat MotivatethisBook xv

1TheLangevinEquationandStochasticProcesses 1

1.1GeneralFramework 1

1.2TheOrnstein-Uhlenbeck(OU)Process 5

1.3TheOverdampedLimit 8

1.4TheOverdampedHarmonicOscillator:AnOrnstein–Uhlenbeck process 11

1.5DifferentialFormandDiscretization 12

1.5.1Euler-MaruyamaDiscretization(EMD)andItôProcesses 15

1.5.2StratonovichDiscretization(SD) 17

1.6RelationBetweenItôandStratonovichIntegrals 19

1.7SpaceVaryingDiffusionConstant 21

1.8ItôvsStratonovich 23

1.9DetailedBalance 23

1.10MemoryKernel 25

1.11TheManyParticleCase 26 References 26

2TheFokker–PlanckEquation 29

2.1TheChapman–KolmogorovEquation 29

2.2TheOverdampedCase 30

2.2.1DerivationoftheSmoluchowski(Fokker–Planck)Equationusingthe Chapman–KolmogorovEquation 30

2.2.2AlternativeDerivationoftheSmoluchowski(Fokker–Planck) Equation 33

2.2.3TheAdjoint(orReverseorBackward)Fokker–PlanckEquation 34

2.3TheUnderdampedCase 34

2.4TheFreeCase 35

2.4.1OverdampedCase 35

2.4.2UnderdampedCase 36

2.5AveragesandObservables 37 References 39

3TheSchrödingerRepresentation 41

3.1TheSchrödingerEquation 41

3.2SpectralRepresentation 43

3.3GroundStateandConvergencetotheBoltzmannDistribution 44 References 47

4DiscreteSystems:TheMasterEquationandKineticMonte Carlo 49

4.1TheMasterEquation 49

4.1.1Discrete-TimeMarkovChains 49

4.1.2Continuous-TimeMarkovChains,MarkovProcesses 51

4.2DetailedBalance 53

4.2.1FinalStateOnly 54

4.2.2InitialStateOnly 54

4.2.3InitialandFinalState 55

4.2.4MetropolisScheme 55

4.2.5Symmetrization 55

4.3KineticMonteCarlo(KMC) 58 References 61

5PathIntegrals 63

5.1TheItôPathIntegral 63

5.2TheStratonovichPathIntegral 66 References 67

6BarrierCrossing 69

6.1FirstPassageTimeandTransitionRate 69

6.1.1AverageMeanFirstPassageTime 71

6.1.2DistributionofFirstPassageTime 73

6.1.3TheFreeParticleCase 74

6.1.4ConservativeForce 75

6.2KramersTransitionTime:AverageandDistribution 77

6.2.1KramersDerivation 78

6.2.2MeanFirstPassageTimeDerivation 80

6.3TransitionPathTime:AverageandDistribution 81

6.3.1TransitionPathTimeDistribution 82

6.3.2MeanTransitionPathTime 84 References 86

7SamplingTransitionPaths 89

7.1DominantPathsandInstantons 92

7.1.1Saddle-PointMethod 92

7.1.2TheEuler-LagrangeEquation:DominantPaths 92

7.1.3SteepestDescentMethod 96

7.1.4GradientDescentMethod 97

7.2PathSampling 98

7.2.1MetropolisScheme 98

7.2.2LangevinScheme 99

7.3BridgeandConditioning 99

7.3.1FreeParticle 102

7.3.2TheOrnstein-UhlenbeckBridge 102

7.3.3ExactDiagonalization 104

7.3.4CumulantExpansion 105 References 111 AppendixA:GaussianVariables 111 AppendixB 113

8TheRateofConformationalChange:Definitionand Computation 117

8.1First-orderChemicalKinetics 117

8.2RateCoefficientsfromMicroscopicDynamics 119

8.2.1ValidityofFirstOrderKinetics 120

8.2.2MappingContinuousTrajectoriesontoDiscreteKineticsand ComputingExactRates 123

8.2.3ComputingtheRateMoreEfficiently 126

8.2.4TransmissionCoefficientandVariationalTransitionState Theory 128

8.2.5HarmonicTransition-StateTheory 129 References 131

9Zwanzig-Caldeiga-LeggettModelforLow-Dimensional Dynamics 133

9.1Low-DimensionalModelsofReactionDynamicsFroma MicroscopicHamiltonian 133

9.2StatisticalPropertiesoftheNoiseandtheFluctuation-dissipation Theorem 137

9.2.1EnsembleApproach 138

9.2.2Single-TrajectoryApproach 139

9.3Time-ReversibilityoftheLangevinEquation 142 References 145

x Contents

10EscapefromaPotentialWellintheCaseofDynamics ObeyingtheGeneralizedLangevinEquation:General SolutionBasedontheZwanzig-Caldeira-Leggett Hamiltonian 147

10.1DerivationoftheEscapeRate 147

10.2TheLimitofKramersTheory 150

10.3SignificanceofMemoryEffects 152

10.4ApplicationsoftheKramersTheorytoChemicalKineticsin CondensedPhases,ParticularlyinBiomolecularSystems 153

10.5ACommentontheUseoftheTerm“FreeEnergy”inApplicationto ChemicalKineticsandEquilibrium 155 References 156

11DiffusiveDynamicsonaMultidimensionalEnergy Landscape 157

11.1GeneralizedLangevinEquationwithExponentialMemorycanbe Derivedfroma2DMarkovModel 157

11.2TheoryofMultidimensionalBarrierCrossing 161

11.3BreakdownoftheLangerTheoryintheCaseofAnisotropic Diffusion:theBerezhkovskii-ZitsermanCase 167 References 171

12QuantumEffectsinChemicalKinetics 173

12.1WhenisaQuantumMechanicalDescriptionNecessary? 173

12.2HowDotheLawsofQuantumMechanicsAffecttheObserved TransitionRates? 174

12.3SemiclassicalApproximationandtheDeepTunnelingRegime 177

12.4PathIntegrals,Ring-PolymerQuantumTransition-StateTheory, InstantonsandCentroids 184 References 191

13ComputerSimulationsofMolecularKinetics: Foundation 193

13.1ComputerSimulations:StatementofGoals 193

13.2TheEmpiricalEnergy 195

13.3MolecularStates 197

13.4MeanFirstPassageTime 199

13.5CoarseVariables 199

13.6Equilibrium,Stable,andMetastableStates 200 References 202

14TheMasterEquationasaModelforTransitionsBetween Macrostates 203 References 211

15DirectCalculationofRateCoefficientswithComputer Simulations 213

15.1ComputerSimulationsofTrajectories 213

15.2CalculatingRatewithTrajectories 219 References 221

16ASimpleNumericalExampleofRateCalculations 223 References 231

17RareEventsandReactionCoordinates 233 References 240

18Celling 241 References 252

19AnExampleoftheUseofCells:AlanineDipeptide 255 References 257 Index 259

Acknowledgments

REacknowledgesresearchsupportfromtheRobertA.WelchFoundation (GrantNo.F-1896)andfromtheNationalInstitutesofHealth(GrantNo. GM059796).Heisalsogratefultohisco-workersAlfredoECardenas,Arman Fathizadeh,PiaoMa,KatelynPoole,ClarkTempleton,andWeiWei,whohave providedmanyconstructivecommentsduringthepreparationofthistext. TheassistanceofArmanFatizadehandAtisMuratinthepreparationofthe coverimageandofAlfredoE.CardenasandWeiWeiinproductionofsomeof thefiguresisgreatlyacknowledged.TheMuellertrajectorieswerecomputed usingaprogramwrittenbyPiaoMaoandthealaninedipeptidecalculations wererunbyWeiWei.

DEMacknowledgesresearchsupportfromtheRobertA.WelchFoundation(GrantNo.F-1514)andtheNationalScienceFoundation(GrantNo.CHE 1566001),aswellastheilluminatingcommentsfromhiscolleaguesAlexanderM.Berezhkovskii,PeterHamm,ErikHolmstrom,HannesJonsson,Eduardo Medina,DanielNettels,EliPollak,RohitSatija,BenjaminSchuler,andFlurin Sturzenegger.

HOwouldliketothankM.BauerandK.Mallickfornumerousilluminating discussions.

Introduction:HistoricalBackgroundandRecent DevelopmentsthatMotivatethisBook

Thisbookgrewfromthelecturesgivenatthesummerschoolsthatthethree ofusorganizedinTellurideandinLausanne.Thepurposeoftheschools wastointroduceyoungresearcherstomodernkinetics,withanemphasisto applicationsinlifesciences,inwhichrigorousmethodsrootedinstatistical physicsareplayinganincreasingrole.Indeed,molecular-levelunderstanding ofvirtuallyanyprocessthatoccursincellularenvironmentrequiresakinetic descriptionaswellastheabilitytomeasureand/orcomputetheassociated timescales.Similarly,dynamicphenomenaoccurringinmaterials,suchas nucleationorfracturegrowth,requireakineticframework.Experimentaland theoreticaldevelopmentsofthepasttwodecades,whicharebrieflyoutlined below,rejuvenatedthewell-establishedfieldofkineticsandplaceditatthe junctureofbiophysics,chemistry,molecularbiology,andmaterialsscience; thecompositionoftheclassattendingtheschools,whichincludedgraduate studentsandpostdocsworkingindiverseareas,reflectedthisrenewedinterest inkineticphenomena.Inthisbookwestrivetointroduceadiverseaudience tothemoderntoolkitofchemicalkinetics.Thistoolkitenablespredictionof kineticphenomenathroughcomputersimulations,aswellasinterpretation ofexperimentalkineticdata;itinvolvesmethodsthatrangefromatomistic simulationstophysicaltheoriesofstochasticphenomenatodataanalysis.

Thematerialpresentedinthisbookcanbelooselydividedintothree interconnectedparts.ThefirstpartconsistsofChapter1–7andprovidesa comprehensiveaccountofstochasticdynamicsbasedonthemodelwherethe dynamicsoftherelevantdegree(s)offreedomisdescribedbytheLangevin equation.TheLangevinmodelisespeciallyimportantforseveralreasons: First,itisthesimplestmodelthatcapturestheessentialfeaturesofanyproper kinetictheory,therebyofferingimportantinsights.Second,itoftenoffers aminimal,low-dimensionaldescriptionofexperimentaldata.Third,many problemsformulatedwithinthismodelcanbesolvedanalytically.TheoreticalapproachestoLangevindynamicsdescribedinthisbookincludethe Fokker–Planckequation,mappingbetweenstochasticdynamicsandquantum mechanics,andPathIntegrals.

xvi Introduction:HistoricalBackgroundandRecentDevelopmentsthatMotivatethisBook

Thesecondpartofthebook,comprisedofChapter8–12,describesratetheoriesandexplainsourcurrentunderstandingofwhata“reactionrate”isand howitisconnectedtotheunderlyingmicroscopicdynamics.Theratetheories discussedinthispartrangefromthesimpletransitionstatetheorytomultidimensionaltheoriesofdiffusivebarriercrossing.Whiletheywereoriginally formulatedindifferentlanguagesandinapplicationtodifferentphenomena, weexplaintheirinterrelationshipandformulatethemusingaunifiedlanguage.

Finally,thethirdpart,formedbyChapter13–19,focusesonatomisticsimulationsanddetailsapproachesthataimto:(i)performsuchsimulationsinthe firstplace,(ii)predictkineticphenomenaoccurringatlongtimetimescalesthat cannotbereachedviabruteforcesimulations,and(iii)obtaininsightsfromthe simulationdata(e.g.byinferringthecomplexdynamicalnetworksortransition pathwaysinlarge-scalebiomolecularrearrangements).

Westartwithabriefhistoricaloverviewofthefield.Tomostchemists,the centerpieceofchemicalkineticsistheArrheniuslaw.Thisphenomenological rulestatesthattheratecoefficientofachemicalreaction,whichquantifieshow fastsomechemicalspeciescalledreactant(s)interconvertintodifferentspecies, theproduct(s),canbewrittenintheform

Tosomeonewithatraditionalphysicsbackground,however,thisrulemay requiremuchexplaining.Inclassicalmechanics,thestateofamolecular systemconsistingof N atomsisdescribedbyitspositioninphasespace (x, p), where x isthe3N -dimensionalvectorcomprisedofthecoordinatesofits atomsand p thecorrespondingvectoroftheirmomenta.Thetimeevolution inphasespaceisgovernedbyNewton’ssecondlaw,whichresultsinasystem of6N differentialequations.Whatexactlydoesachemistmeanby“reactants” or“products”?When,why,andhowcanthecomplexNewtoniandynamics (orthedynamicsgovernedbythelawsofquantummechanics)bereducedto asimpleratelaw,whatisthephysicalmeaningof �� and Ea ,andhowcanwe predicttheirvalues?

EarlysystematicattemptstoanswerthesequestionsstartedintheearlytwentiethcenturywithworkbyReneMarcelin[1]andculminatedwiththeideasof transitionstatetheoryformulatedbyWigner[2],Eyring[3]andothers(see,e.g., thereview[4].Inthemodernlanguage(closesttoWigner’sformulation)“transitionstate”isahypersurfacethatdividesthephasespaceintothereactants andproducts;itfurtherpossessesthehypotheticpropertythatanytrajectory crossingthishypersurfaceheadingfromthereactantstotheproductswillnot turnback.Interestingly,theexistence(inprinciple)ofsuchahypersurfacehas remainedanunsettledissueevenintherecentpast[5].Inpractice,transition statetheoryisoftenagoodapproximationforgas-phasereactionsinvolving fewatoms,orfortransitionsinsolids,whichhavehighsymmetry.Butpractical

Introduction:HistoricalBackgroundandRecentDevelopmentsthatMotivatethisBook

transition-statetheoryestimatesoftheprefactor �� forbiochemicalphenomena thattakeplaceinsolutionareusuallyoffbymanyordersofmagnitude.

SeveralimportantdevelopmentsadvancedourunderstandingoftheArrheniuslawlaterinthetwentiethcentury.First,Kramersinhisseminalpaper[6] approachedtheproblemfromadifferentstartingpoint:insteadofconsideringthedynamicsinthehigh-dimensionalspaceinvolvingallthedegreesof freedom,heproposedamodelwhereoneimportantdegreeoffreedomcharacterizingthereactionistreatedexplicitly,whiletheeffectoftheremaining degreesoffreedomiscapturedphenomenologicallyusingtheoryofBrownian motion.Kramers’ssolutionofthisproblemdescribedinChapter6ofthisbook hashadatremendousimpact,particularly,onbiophysics.

Second,Keckandcollaborators[7]showedthat,evenifthehypersurfaceseparatingtheproductsfromthereactantsdoesnotprovideapointofnoreturn andcanberecrossedbymoleculartrajectories,itisstillpossibletocorrectfor suchrecrossings,and–iftherearenottoomany–practicalcalculationmethodsexistthatwillyieldtheexactrate[8–10].

Third,Kramers’sideaswereextendedbeyondsimple,one-dimensionalBrowniandynamicstoincludemultidimensionaleffects[11]andconformational memory[12,13].Moreover,aunificationofvariousratetheorieswasachieved [4,14,15]usingtheideathatBrowniandynamicscanbeobtainedfromthe conservativedynamicsofanextendedsystemwherethedegreeoffreedomof interestiscoupledbilinearlytoacontinuumofharmonicoscillators[16,17]. ThisunifiedperspectiveonratetheorieswillbeexploredindetailinChapter 9–11ofthisbook.

Thereis,however,moretokineticsthancalculatingtherateofanelementary chemicalstep.Manybiophysicaltransportphenomenacannotbecharacterized byasingleratecoefficientasinEq.(1).Yetadescriptionofsuchphenomenainfullatomisticdetailwouldbothlackinsight(i.e.failtoprovidetheir salientfeaturesormechanisms)andbeprohibitivecomputationally.Thechallengeistofindamiddlegroundbetweensimplephenomenologicaltheories andexpensivemolecularsimulations,ataskthathasbeentackledbyahostof newmethodsthathaveemergedinthelastdecadeandthatemploy“celling” strategiesdescribedinChapters18–19ofthisbook.

Anotherrecentdevelopmentinmolecularkineticshasbeendrivenby single-moleculeexperiments,whosetimeresolutionhasbeensteadilyimproving:ithasbecomepossibletoobservepropertiesofmoleculartransitionpaths bycatchingmolecules enroute fromreactantstoproducts[18,19]asthey crossactivationbarriers.Suchmeasurementsprovidecriticaltestsofvarious ratetheoriesdescribedinChapters6,10,11,andinformusaboutelusive reactionmechanisms.Propertiesoftransitionpaths,suchastheirtemporal durationanddominantshapearediscussedinChapter6–7.

Giventhatthefocusofthisbookisonthedynamicphenomenaincondensed phases,andespeciallyonbiophysicalapplications,anumberoftopicswere

xviii Introduction:HistoricalBackgroundandRecentDevelopmentsthatMotivatethisBook leftout.Those,forexample,includetheoriesofreactionrateinthegasphase. Likewise,thelow-frictionregime(alsoknownastheenergydiffusionregime) andtherelatedKramersturnoverproblem[15]forbarriercrossingarenot coveredhere,asthosearerarelypertinentforchemicaldynamicsinsolution. Throughoutmostofthisbook,itwasassumedthatthedynamicsofmolecules canbedescribedbythelawsofNewtonianmechanics,withquantummechanicsonlyenteringimplicitlythroughthegoverninginter-andintramolecular interactions.Thisassumptionholdsinmanycases,butthelawsofquantum mechanicsbecomeimportantwhenachemicalreactioninvolvesthetransfer ofalightparticle,suchasprotonorelectron,and/orwhenitoccursatavery lowtemperature.Thesubjectofquantumratetheorycouldfillaseparatebook [20,21];hereitsdiscussionislimitedtoasinglechapter(Chapter12)thatis onlymeanttoprovidearudimentaryintroduction.

References

1 Marcelin,R.(1915). Ann.Phys. 3:120.

2 Wigner,E.(1932). Z.Phys.Chem.Abt.B 19:203.

3 Eyring,H.(1935).Theactivatedcomplexinchemicalreactions. J.Chem. Phys. 3:107.

4 Pollak,E.andTalkner,P.(2005).Reactionratetheory:whatitwas,whereis ittoday,andwhereisitgoing? Chaos 15(2):26116.

5 Mullen,R.G.,Shea,J.E.,andPeters,B.(2014).Communication:anexistence testfordividingsurfaceswithoutrecrossing. J.Chem.Phys. 140(4):041104.

6 Kramers,H.A.(1940).Brownianmotioninafieldofforceandthediffusion modelofchemicalreactions. Physica 7:284–304.

7 Shui,V.H.,Appleton,J.P.,andKeck,J.C.(1972).MonteCarlotrajectorycalculationsofthedissociationofHClinAr. J.Chem.Phys. 56:4266.

8 Chandler,D.(1978). J.Chem.Phys. 68:2959.

9 Bennett,C.H.(1977).Moleculardynamicsandtransitionstatetheory:the simulationofinfrequentevents.In: AlgorithmsforChemicalComputations, vol.46,63–97.AmericanChemicalSociety.

10 Truhlar,D.G.,Garrett,B.C.,andKlippenstein,S.J.(1996).Currentstatusof transition-statetheory. J.Phys.Chem. 100(31):12771–12800.

11 Langer,J.S.(1969). Ann.Phys.(N.Y.)54:258.

12 Grote,R.F.andHynes,J.T.(1980).Thestablestatespictureofchemical reactions.II.Rateconstantsforcondensedandgasphasereactionmodels. J.Chem.Phys. 73(6):2715–2732.

13 Grote,R.F.andHynes,J.T.(1981).Reactivemodesincondensedphasereactions. J.Chem.Phys. 74(8):4465–4475.

14 Pollak,E.(1986).Theoryofactivatedrate-processes-anewderivationof Kramersexpression. J.Chem.Phys. 85(2):865–867.

Introduction:HistoricalBackgroundandRecentDevelopmentsthatMotivatethisBook

15 Hanggi,P.,Talkner,P.,andBorkovec,M.(1990).50yearsafterKramers. Rev.Mod.Phys. 62:251.

16 Zwanzig,R.(2001). NonequilibriumStatisticalMechanics.OxfordUniversity Press.

17 Caldeira,A.O.andLeggett,A.J.(1983).Quantumtunnelinginadissipative system. Ann.Phys. 149:374.

18 Chung,H.S.andEaton,W.A.(2018).Proteinfoldingtransitionpathtimes fromsinglemoleculeFRET. Curr.Opin.Struct.Biol. 48:30–39.

19 Neupane,K.,Foster,D.A.,Dee,D.R.etal.(2016).Directobservationof transitionpathsduringthefoldingofproteinsandnucleicacids. Science 352 (6282):239–242.

20 Benderskii,V.A.,Makarov,D.E.,andWight,C.A.(1994). ChemicalDynamicsatLowTemperatures.NewYork:Wiley.

21 Nitzan,A.(2006).ChemicalDynamicsinCondensedPhases:Relaxation, TransferandReactionsinCondensedMolecularSystems.Oxford/New York:OxfordUniversityPress;pxxii,719p.

TheLangevinEquationandStochasticProcesses

1.1GeneralFramework

Inthissection,wewilldiscusshowtodescribeasystemofinteractingparticles incontactwithaheatbath(orthermalreservoir)attemperature T .Examplesof suchsystemsarecountless.Justtociteafew,wewilloftenrefertoaproteinina solvent,thebindingofaproteintoaDNA,etc.Ourapproachcloselyfollowsthe originalworkofPaulLangevin[1].Verycompletepresentationsofthesubjects treatedinthispartcanbefoundinthefollowingbooks[2].Forathorough reviewofthemathematicsofstochasticprocesses,wereferthereadertothe bookbyGardiner[3].

Considerasingleparticleofmass m inafluid.Inthefollowing,wewill assumethephysicalspacetobeone-dimensional,sincegeneralizationto arbitrarydimensionisstraightforward.Theformalismdevelopedheremay applytosome reactioncoordinates ofthesystem.Reactioncoordinatesare low-dimensionalcollectivevariablesthatrepresenttheevolutionofasystem alongatransformationpathway(physicalorchemical).

Goingbacktotheone-dimensionalcase,inthefluid,theparticleissubject toafrictionforce Ff givenby Stokeslaw [4]

where �� isthevelocityoftheparticleand �� isthefrictioncoefficient.The Newtonequation fortheparticleisthus

whichcanbeintegratedas

where ��(0) istheinitialvelocityoftheparticle.Inthelimitoflargetime t → ∞, ��(t ) → 0.Duetothefrictionforce,theparticlegoestorestatlargetime,with relaxationtime ��r = m∕��

MolecularKineticsinCondensedPhases:Theory,Simulation,andAnalysis, FirstEdition.RonElber,DmitriiE.MakarovandHenriOrland. ©2020JohnWiley&SonsLtd.Published2020byJohnWiley&SonsLtd.

1TheLangevinEquationandStochasticProcesses

However,atlargetime,theparticleshouldreachthermalequilibriumwith theheatbathattemperature T ,whichimpliesthattheprobabilitydistribution foritsvelocityshouldbethe Maxwelldistribution

where �� = 1∕kB T and kB istheBoltzmannconstant.Aconsequenceofthe Maxwelldistributionisthe equipartitiontheorem,whichstatesthatatthermal equilibrium

ThisequipartitiontheoremisjustaconsequenceoftheGaussianintegral

ThisisobviouslynotcompatiblewithEq.(1.3)whichimpliesavanishing velocityatlongtime.Toovercomethisdifficulty,Langevinproposedtoadda time-dependentrandomforce �� (t ) tothoseactingontheparticle,toaccount forthethermalagitation(Brownianmotion)andforthecollisionswiththe heatbathmoleculesattemperature T .TheNewtonequationthusbecomes

Therandomforce �� (t ) isalsocalled randomnoise.Theaverageofthisforce shouldvanish,asitrepresentsrandomthermalagitationwithnospecificdirectionorintensity

⟨�� (t )⟩ = 0(1.8)

forany t .Thebracketmeansanaverageoverallpossiblenoisehistories,thatis overalargenumberofevolutionsofthesameparticle,withdifferentrealizationsofthenoise.

TakingtheaverageofEq.(1.7)overdifferentrandomnoises,wehave

whichisthesameasEq.(1.2)aboveandimplies ⟨��(+∞)⟩ = 0atlargetime.The integrationofEq.(1.7)isstraightforwardandyields

whichatlargetime t → ∞

Physically,therandomforce �� (t ) isduetocollisionsoftheheatbathmolecules withthestudiedparticle.Thereforetheintensityanddirectionoftherandom forcechangesateachcollision,andthusthecorrelationtime ��c oftherandom forceisoftheorderofmagnitudeofthetypicaltimespanbetweencollisions oftheheatbathmoleculeswiththeparticle.Iftherelaxationtime ��r ofthe particleandthetimescales t atwhichwestudytheparticleismuchlargerthan thecorrelationtime ��c ,wemayneglectthetimecorrelationsoftherandom forceandwrite

where �� (t t ′ ) denotestheDiracdeltafunction(ordistribution),whichisnon zeroonlyif t = t ′ .Wenowshowhowtoadjustthecoefficient �� inorderto satisfytheequipartitiontheorematthermalequilibrium(1.5).Thevarianceof thevelocityisgivenby

Atlargetime,wehave ⟨

,whichfrom(1.5)implies

Wethusseethatthecoefficient �� isrelatedtothefrictioncoefficient �� through therelation

Wenowturntothegeneralcaseofaparticleinaheatbathattemperature T onwhichaforce F (x, t ) (possiblytime-dependent)isacting.Again,addingthe Stokesforce,theNewtonequationreads

Iftheforceisconservative,itisthegradientofapotentialenergy F =− dU dx and wehave

1TheLangevinEquationandStochasticProcesses

Multiplyingthisequationbythevelocity �� = dx dt m�� d �� dt + ����2 + �� dU dx = d dt ( 1 2 m��2 + U ) + ����2

where E = 1 2 m��2 + U isthetotalenergyofthesystem.Intheaboveequation,we haveusedthefactthatforatimeindependentpotential,

�� dU (x(t ))

dx .Equation(1.18)implies dE dt =−����2 ≤ 0andthereforetheenergyofthe systemdecreaseswithtime.When t → +∞,iftheenergyisboundbelow,the systemwillfallintoalocalminimumofthepotentialenergy U (x),withavanishingvelocity.Inthisstate, dU dx = 0and �� = 0andEq.(1.17)issatisfied.

Asbefore,theequipartitiontheoremisnotsatisfied,andtoovercomethis problem,weagainintroducearandomforce �� (t ) andobtainthe Langevin equation

Theseequationsdonotspecifyhighermomentsoftherandomforce.In thefollowing,wewillassumethattheforceis Gaussiandistributed ,with correlationfunctionsgivenby(1.20).Aparticlesubjecttosucharandomforce iscalleda Brownianparticle.Inthisequation,theterm �� dx dt termisthefriction term,theforceterm F iscalledthe driftterm andthenoiseterm �� (t ) iscalled the diffusionterm.

Equation(1.19)iscalleda stochasticdifferentialequation (SDE)andconstitutesaveryactivebranchofmodernmathematics.Thisequationdescribesa Markovprocess,thatisastochasticprocesssuchthatthestateofthesystemat time t + dt depends(stochastically)onlyonthestateofthesystemattime t . ArandomnoisesuchasinEq.(1.20)withconstantvarianceandzerocorrelationtimeiscalleda whitenoise.Anoisewithnon-zerocorrelationtimeis called colorednoise andwewillseeanexampleofthatinsection1.2onthe Ornstein-Uhlenbeckprocess.Tobecomplete,letusmentionthatwehavediscussedonlythecasewhenthefrictiontermisinstantaneous.Insomesystems, thefrictionforcedoesnottaketheinstantaneousStokesform,butmaydepend ontheprevioushistoryofthesystem.Insection1.8,wewillbrieflydescribe thecasewhenthefrictionforceisexpressedintermsofamemorykernel.We willalsodiscussthecasewhentheparticlemovesinainhomogeneoussolvent, wherethediffusioncoefficient D dependson x.

1.2TheOrnstein-Uhlenbeck(OU)Process

Followingtheprevioussection,weconsideraparticleintheabsenceofanexternalforce F .Eq.(1.19)becomes

wherethenoiseisGaussian,withmomentsgivenbyEq.(1.20).Theabove equationcanbeexpressedinasimplerformbywritingitintermsofthe velocity

Thestochasticmotionoftheparticleiscalledan Ornstein-Uhlenbeckprocess (OU).ThesolutionofEq.(1.22)isgivenby

where ��0 istheinitialvelocityoftheparticleand �� (t ) isawhiteGaussiannoise withcorrelationsgivenbyEq.(1.20).Byintegratingthisequationweget

(t )= x0 + ��

where x0 istheinitialpositionoftheparticle.Letuscalculatethecorrelation timeofthevelocityoftheparticle(inabsenceofanyexternalforce).Theaverage velocityisgivenby

sincetheaveragenoisevanishes(fromEq.(1.20)).Thevelocitycorrelation functionisgivenby

whereweusethenotation ⟨…⟩c todenotethe connected correlationfunctionof thevariable,alsocalledthe secondcumulant ,wheretheaverageofthevariable hasbeensubtracted.Intheaboveequation,the �� functionisnon-zeroonly

1TheLangevinEquationandStochasticProcesses

when �� = �� ′ , andthiscanbesatisfiedifandonlyifboth �� and �� ′ aresmaller thanthesmallestof t and t ′ . Letusdenote

t< = inf (t , t ′ )

t> = sup(t , t ′ )

whereinf (t , t ′ ) andsup(t , t ′ ) denotethesmaller(resp.larger)ofthetwotimes t and t ′ .Then t + t ′ = t< + t> andwecanwrite ⟨��(t )��(t ′ )⟩c =

Atlargetimes, t , t ′ → ∞,thecorrelationfunctiondecaysexponentiallywith |t t ′ |

WethusseethatthevelocityofanOUprocessisalsoaGaussiannoise,but sinceithasafinitecorrelationtime,itisanexampleofacolouredGaussian noise.Bysetting t = t ′ inEq.(1.27),weobtainthevarianceofthevelocityas

Atlargetime,werecovertheequipartitiontheorem.Wenotethatthevariance ofthevelocityisfinite,andhasafinitelimitwhen t → ∞.

Duetoitsinertialmass m,aBrownianparticlehasatendencytocontinue inthedirectionofitsvelocity,despitetheactionoftherandomforce.Asa result,thevelocityoftheparticlehasafinitecorrelationtime,thatis,atime overwhichitsdirectionandlengthpersist.FromEq.(1.28),thecorrelationtime ofthevelocityis

whichisthetime-scaleoverwhichthevelocitypersists.Beyondthattime,the velocitiesoftheparticlescanbeconsideredasindependentrandomvariables. Aswewillshownext,thisimpliesthatifweareinterestedinthebehaviorof thesystemattimescaleslargerthan m∕�� ,wecanneglectthemassterminthe Langevinequation.

Itisinterestingtocomputetheprobabilitydistributionofthevelocityand ofthepositionoftheparticle.Thereisapowerfultheoreminmathematics

1.2TheOrnstein-Uhlenbeck(OU)Process 7

whichstatesthatifarandomvariableisalinearcombinationofsomeGaussianrandomvariables,thenitisitselfaGaussiandistributedrandomvariable, anditsdistributionisthusentirelyspecifiedbyitsaverageanditsvariance(see AppendixA).FromEq.(1.23),weseethatthevelocityisalinearcombination oftheGaussiannoise �� (t ) andthusitisGaussiandistributedwithdistribution

where �� isthenormalizationconstant.Itfollowsthatthevelocityoftheparticleisalwaysbound,anditsdistributionconvergestotheMaxwelldistribution. Similarly,theaverageandvarianceofthepositioncanbecomputed.Theaverageisgivenby

andthevarianceis

Again,since ��2 mustbeequalto ��4 ,theyarebothlimitedbythesmallestofthe twotimes ��1 and ��3 .Wedenote

��< = inf (��1 ,��3 )

��> = sup(��1 ,��3 )

Withthisnotation,Eq.(1.33)becomes

⟨(x(t )− ⟨

where D = kB T �� isaconstant.Atlargetime,werecoverthestandarddiffusion law

1TheLangevinEquationandStochasticProcesses

where (Δx)2 = ⟨(x(t )− ⟨x(t )⟩)2 ⟩.Eq.(1.35)impliesthat D isthediffusioncoefficientoftheparticleandwehavethe Einsteinrelation D = kB T �� (1.36)

whichrelatesthediffusionandthefrictioncoefficients[5].Atshorttimes t ≪ m �� ,wemayexpandEq.(1.34)tosecondorderin t andweobtain (Δx)2 ∼ D�� m t 2

whichshowsthatthemovementoftheparticleisballisticatshorttimes. Knowingtheaverageandvarianceoftheposition,wecancalculatetheprobabilitydistributionoftheposition.Forlargetime,weobtain

Duetotheinertiaoftheparticle,toitsinitialvelocity ��0 andtothefriction force,theparticleremainscenteredaround x0 + ��0 �� m anddiffusesaroundthat position.

1.3TheOverdampedLimit

Fromtheprecedingsection,itappearsthatifweareinterestedintimescales largerthanthecorrelationtime ���� = m∕�� ,thevelocitiescanbeconsideredas randomindependentvariables.Aswewillnowshow,thisamountstoneglect themasstermintheLangevinequation.Indeed,takingtheFouriertransform oftheLangevinEquation(1.19),wehave

Ifweareinterestedinthelongtimelimitofthesystem,onlythesmall �� Fourier componentsbecomerelevant.Forsmallenough ��,themasstermcarriesan ��2

1.3TheOverdampedLimit 9

termandcanbeneglectedcomparedtothefrictiontermwhichislinearin ��. TheLangevinequationthusreducesto

withthenoisesatisfyingequationsEq.(1.20).Thisequationiscalledthe overdampedLangevinequation,or Browniandynamics (BD).Dividingthis equationby �� andusingEq.(1.36),thisequationisoftenwrittenas

Thetimescalebeyondwhichthemasstermisirrelevantcanbereadoff Eq.(1.39).Indeed,for �� sufficientlysmall,wehave

whichinrealtimetranslatesinto

Therefore,fortimescaleslargerthan ���� = m �� ,whichturnsouttobethe correlationtimeofvelocitiesfromEq.(1.30),onecanneglectthemassterm intheLangevinEquation(1.19)andreplaceitbytheoverdampedLangevin Equation(1.41).Letusshowthatintheoverdampedlimit,thediffusion relationEq.(1.35)issatisfied.Indeed,inthecaseofnopotential U = 0,wecan integrateEq.(1.41)andobtain

1TheLangevinEquationandStochasticProcesses

wherewehaveassumedthattheparticleisatpoint0attime0.Therefore,using Eq.(1.20),thecorrelationfunctionofthepositionisgivenby

andthesquaredisplacementisgivenby

WethusrecovertheEinsteinrelation(1.36).Itfollowsthattheshorttimescale ���� beyondwhichtheoverdampedlimitisvalidisgivenby

where M isthemolarmassofthesolute,and R ≃ 8.31J/mol/Kistheidealgas constant.Forexample,foranamino-acidinwaterwithaveragemolarmass M = 110g/mol,anddiffusioncoefficientatroomtemperaturetypically10 5 cm2 ∕s weobtain ��s = 5.10 14 s.Althoughtypicallyinmoleculardynamicssimulations, thediscretizationtimestepisoftheorderoffemtoseconds(10 15 s),inmany instancesoneisnotinterestedindetailsofthedynamicsattimescalesbelow 10 13 s,andthusBrowniandynamicswillbeappropriate.

Tobecomplete,letusnotethatforasystemwith N degreesoffreedom(e.g. amany-particlesystemin3d),theLangevinEquationtakestheform

wherethediffusionandfrictioncoefficients Di ,��i with Di = kB T

maydependonthenatureofthedegreeoffreedom i.

1.4TheOverdampedHarmonicOscillator:AnOrnstein–Uhlenbeckprocess 11

1.4TheOverdampedHarmonicOscillator:An Ornstein–Uhlenbeckprocess

Whenstudyingthemotionofasystemnearalocalminimum,itisreasonable toexpandthepotentialtosecondorderarounditandapproximateitbyaharmonicpotential.Consideraparticleinaharmonicpotential

withspringconstant K .TheoverdampedLangevinequationis

WeseethatthisequationisformallyequivalenttoEq.(1.22),andthustheoverdampedmotionofaparticleinaharmonicpotentialisanOUprocess.Itfollows thatalltheresultsofsection1.2canbeadapted.Defining �� = D�� K ,weobtain

whichshowsthatthemotionoftheparticleremainsbound,withacorrelation timeequalto

FromEq.(1.56),weseethattherelaxationtime ��r ofthesystemtoequilibrium isalsogivenby

Thep.d.foftheparticleisGaussianatalltime,givenby

andconvergestotheBoltzmanndistributionwhen t → ∞.InFigure1.1we showtheplotofatypicalOUtrajectoryinthepotential U = x2 ∕2attemperature T = 0.1withtimestep dt = 0.001andtotalduration t = 50.Ascanbeseen, themotionof x isconfinedaround x = 0.

Figure1.1 Ornstein-Uhlenbecktrajectory.

1.5DifferentialFormandDiscretization

Inmostofthefollowing,wewillsticktotheoverdampedLangevinequation, orBrownianDynamics(BD).Indifferentialform,theoverdampedLangevin equation(1.41)canbewrittenas

where dB(t )= �� (t )dt isthedifferentialofaBrownianprocess B(t ).ABrownianprocessisacontinuousstochasticprocess,whosevariationsduringany infinitesimaltimeinterval dt areindependentnormalGaussianvariablesof zeromeanandvarianceproportionalto dt .AccordingtoEq.(1.44),wehave

where �� (0, 1) isaGaussiannormalvariableof0meanandunitvariance.We canwritesymbolically

Thisequationdescribesan Itôdiffusion,andhasbeenthoroughlystudiedin mathematics.Equations(1.60)and(1.61)showthefundamentalresultthatthe typicaldisplacementoftheparticleduringtime dt isoforder √dt

Thevelocityoftheparticle dx dt ∼ 1 √dt isdiscontinuousatalltime.Infact,since thedriftterm F �� dt isoforder dt ,takingthesquareofEq.(1.60)andkeeping termstoorder dt yields

Similarly,thefullLangevinEq.(1.19)canbewrittenindifferentialformusing themomentum p = m dx

Inthatcase,Eq.(1.63)isreplacedby

andtheaccelerationoftheparticleisdiscontinuousatalltime.

ThereisaninfinitenumberofwaystodiscretizeEq.(1.60)or(1.65),atorder dt . Atheoremonstochasticprocessesstatesthatthecontinuouslimitofthe Itôdiffusionexists,andisindependentofthediscretizationoftheequation. Wedefinethetimestep dt andthesequenceoftimes

IntegratingEq.(1.60)between ��n and ��n+1 = ��n + dt ,wehave

where xn = x(��n ) and xn+1 = x(��n+1 ).Thestochastictermin(1.70)isasum(integral)ofindependentGaussianrandomvariables dB(s).ItisthusaGaussian randomvariablewith

Wehave

1TheLangevinEquationandStochasticProcesses

If m ≠ n,thetwointegrationintervalsaredisjointandthe �� functionvanishes. Therefore

Thislastequationshowsthatthenoise ��n isoforder √dt .Tomakethisscalingexplicit,weintroducethevariables ��n = ��n √2Ddt whicharenormalGaussian randomvariableswith0mean,unitvariance

andprobabilitydistribution

Theintegraloftheforce F (x(s), s) overtheinterval [��n ,��n+1 ] isoforder ��n+1 ��n = dt .Wemaythuswriteitas

where Φn dependsonthetrajectory x(s) betweentimes ��n and ��n+1 . Equation(1.70)takestheform

ThevariousdiscretizationschemesoftheLangevinequationcorrespondtodifferentexpressionsfor Φn ,whicharediscussedbelow.

AcrucialpointtonoteinEq.(1.81)isthattheforceterm Φn ,alsocalledthe driftterm,isoforder dt sinceitismultipliedby dt ,whiletherandomforce term,alsocalledthediffusionterm,isoforder √dt .Forsmalltimestep,the diffusiontermisthusalwaysmuchlargerthanthedrifttermandtodominant orderin dt ,wehave

⟨(xn+1 xn )2 ⟩ ∼ 2Ddt (1.82)

ItisalsoclearfromthisequationthatthemotionoftheparticleisaMarkov process,sinceitsposition xn+1 attime ��n+1 = ��n + dt dependsstochastically(through ��n )onthestate xn attime ��n only,andnotontheprevious ones.Anothercrucialremarkisthat xn (attime ��n )dependsonall ��m with

m = 0, 1,..., n 1butnotonthenoise ��m′ for m′ = n, n + 1,...,thatis,noton therandomforceattimeslargerorequalto ��n .This causalityprinciple implies thatforanyfunction f ,wehave

Inthecontinuouslimit,thisisequivalentto

Toconcludethissection,wementiontheimportantresultthat

Thisresultistobeunderstoodintermsofstochasticintegrals.Indeed,aswe showinAppendixB,foranyfunction f (x),wehave

Takingaderivativeoftheaboveequationwithrespectto t ,wehave

whichimpliesthesymbolicidentity(1.85).

1.5.1Euler-MaruyamaDiscretization(EMD)andItôProcesses

ThemostcommondiscretizationschemeofEq.(1.41,1.60)istheEulerMaruyamamethod.ItamountstoapproximatetheintegralofEq.(1.80)by

for dt small.TheEuler-Maruyamadiscretizationof(1.60)usesthevaluesofthe functionsattheearliertime ��n .Sinceindifferentialcalculus,allquantitiesare computedtoorder dt ,thetimeatwhichthefunctionisevaluatedinEq.(1.88)is irrelevant,asanyothertimeintheinterval [��n ,��n+1 ] wouldjustaddcorrections ofhigherorder.Withthisscheme,Eq.(1.81)canbewrittenas

Agreatadvantageofthisdiscretizationisthatitprovidesan explicit expression of xn+1 asafunctionof xn .