Brownian Dynamics

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1、Chapter6BrownianDynamicsContents6.1DiscretizationofTime.............................916.2MonteCarloIntegrationofStochasticProcesses.............936.3ItoCalculusandBrownianDynamics....................956.4FreeDiusion...................................966.5Re ectiveBoundaryConditio

2、ns........................100In1827thebotanistRobertBrownexaminedunderhismicrospcopepollenofplantssuspendedinwater.Heobservedthatthepollen,about106mindiameter,performedstochasticmotionsoftheorderof105m.EventhoughBrowncouldnotexplainthecauseofthismotionanycontinuousstochasticproc

3、esslikethatofapollenisnowreferredtoasBrowniandynamics.InthischapterweintroduceanumericaltechniquethatgeneratesasolutionoftheFokker-Planckequation(5.1)bysimulatinganensembleofstochasticprocesses.DuetothesestochasticprocessesonecallsthisnumericalmethodBrowniandynamicsaswell.Weprovid

4、etwoderivationsofthenumericalmethodofBrowniandynamics.IntherstsectionwetransformtheFokker-Planckequation(5.1)intoamulti-dimensionalintegral.Wethenexplaininthesecondsectionhowtoevaluatethismulti-dimensionalintegralusingtheMonteCarlointegrationmethodintroducedinthepreviouschapter.W

5、eshowinthethirdsectionthatthisMonteCarlointegrationisequivalenttosimulatinganensembleofstochasticprocesses.TheequivalenceisshownbyderivingtheBrowniandynamicsmethodasecondtimestartingwiththestochasticdierentialequation(5.2).WeexamplifytheideaofBrowniandynamicsbyapplyingittoafreedi

6、usionmodelinthefourthsectionandconcludethischapterinthefthsectionbyshowinghowtoincorporateboundaryconditionsinBrowniandynamics.6.1DiscretizationofTimeDiscretizationisthebasisofmanynumericalprocedures.ThisalsoholdstrueforBrowniandy-namics.Theobjectofdiscretizationisthecontinuoust

7、imeaxis.Acomputercannotrepresentacontinuouslineorfunction.Theinnitelymanypointswouldsimplynottintoanitedigitalcomputernorcouldtheybeprocessedinaniteamountoftime.Hence,oneneedstoapproximateacontinuousdimensionwithanitesetofpoints.Thisapproximationiscalleddiscretization.9192Bro

8、wnianDynamicsWesolvetheFokker-Pla