Abstract :
Manymultiphaseflowsystemsinvolvecomplexgeometryand/orchaoticflows.Forthisreason,exactsolutionstoinitialvalueproblems(giving,forexample,theexactinitialpositionsofallbubblesflowinginasystem)containbothtoomuchinformationandtoolittleinformation,inasensethatwillbemadeclear.Formanypurposes,flowdetails(suchasripplesonbubbles)canbeneglected,ortheiroveralleffectneedbeconsidered.Informationaboutsuchdetailsisextraneous.Forsomesystems,theboundaryconditionsandinitialconditionscannotbecontrolledsufficientlytoallowrepeatableexperiments.Acorrectdetailedpredictionforsuchasystemcannotbemadebecausetheinitialandboundaryconditionscannotbespecified.Moreover,predictionsforsuchasystemcannotbecomparedindetailtotheexperiment.Consequently,predictionsforaverageshavebecomethestandardforcomplexflowsystems.Inthispaper,theaveragingprocessiselucidatedforsituationswheretheflowissimple.Thecentralconceptistheevolutionofanappropriateprobabilitydensityfunction.Theevolution(rateofchange)oftheprobabilitydensityfunctionisstudiedforlinear,logistic,andLorenzdynamics.Webuildtheconceptsandsolutionstowardtheproblemofturbulentdispersionofparticles.Theeffectofuncertaintyintheinitialconditions,andthedispersionbyrandomflowisdescribedfortheevolutionofaswarmofparticles.
