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Generalized Boltzmann Physical Kinetics BORIS V. ALEXEEV ELSEVIER Generalized Boltzmann Physical Kinetics This page intentionally left blank Generalized Boltzmann Physical Kinetics BORIS V. ALEXEEV MoscowFineChemical TechnologyInstitute Moscow119571,Russia 2004 Amsterdam–Boston–Heidelberg–London–NewYork–Oxford Paris–SanDiego–SanFrancisco–Singapore–Sydney–Tokyo ELSEVIERB.V. ELSEVIERInc. ELSEVIERLtd ELSEVIERLtd SaraBurgerhartstraat25 525BStreet,Suite1900 TheBoulevard,LangfordLane 84TheobaldsRoad P.O.Box211,1000AE SanDiego,CA92101-4495 Kidlington,OxfordOX51GB LondonWC1X8RR Amsterdam,TheNetherlands USA UK UK ©2004ElsevierB.V.Allrightsreserved. ThisworkisprotectedundercopyrightbyElsevierB.V.,andthefollowingtermsandconditionsapplytoitsuse: Photocopying Singlephotocopies of single chapters may be made for personal use as allowed by national copyright laws. 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Notice NoresponsibilityisassumedbythePublisherforanyinjuryand/ordamagetopersonsorpropertyasamatter ofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products,instruc- tionsorideascontainedinthematerialherein.Becauseofrapidadvancesinthemedicalsciences,inparticular, independentverificationofdiagnosesanddrugdosagesshouldbemade. Firstedition2004 LibraryofCongressCataloginginPublicationData AcatalogrecordisavailablefromtheLibraryofCongress. BritishLibraryCataloguinginPublicationData AcataloguerecordisavailablefromtheBritishLibrary. ISBN:0-444-51582-8 ThepaperusedinthispublicationmeetstherequirementsofANSI/NISOZ39.48-1992(PermanenceofPaper). PrintedinTheNetherlands. Contents Preface vii HistoricalIntroductionandtheProblemFormulation 1 Chapter1. GeneralizedBoltzmannEquation 21 1.1. Mathematicalintroduction.Methodofmanyscales 21 1.2. HierarchyofBogolubovkineticequations 33 1.3. DerivationofthegeneralizedBoltzmannequation 39 1.4. GeneralizedBoltzmannH-theoremandtheproblemofirreversibilityoftime 61 1.5. GeneralizedBoltzmannequationanditerativeconstructionofhigher-orderequationsintheBoltzmann kinetictheory 79 1.6. GeneralizedBoltzmannequationandthetheoryofnon-localkineticequationswithtimedelay 83 Chapter2. TheoryofGeneralizedHydrodynamicEquations 91 2.1. Transportofmolecularcharacteristics 91 2.2. HydrodynamicEnskogequations 94 2.3. TransformationsofthegeneralizedBoltzmannequation 96 2.4. Generalizedcontinuityequation 98 2.5. Generalizedmomentumequationforcomponent 101 2.6. Generalizedenergyequationforcomponent 106 2.7. GeneralizedhydrodynamicEulerequations 112 2.8. Boundaryconditionsinthetheoryofthegeneralizedhydrodynamicequations 124 Chapter3. StrictTheoryofTurbulenceandSomeApplicationsoftheGeneralized HydrodynamicTheory 133 3.1. Aboutprinciplesofclassicaltheoryofturbulentflows 133 3.2. TheoryofturbulenceandgeneralizedEulerequations 136 3.3. TheoryofturbulenceandthegeneralizedEnskogequations 149 3.4. Generalizedhydrodynamicequationsandquantummechanics 153 v vi GeneralizedBoltzmannPhysicalKinetics Chapter4. PhysicsofaWeaklyIonizedGas 161 4.1. Relaxationofchargedparticlesin“Maxwellian”gasandthehydrodynamicaspectsofthetheory 161 4.2. Distributionfunctionofthechargedparticlesinthe“Lorentz”gas 168 4.3. Chargedparticlesinalternatingelectricfield 176 4.4. Conductivityofaweaklyionizedgasincrossedelectricandmagneticfields 179 Chapter5. KineticCoefficientsintheTheoryoftheGeneralizedKinetic Equations 187 5.1. LinearizationofthegeneralizedBoltzmannequation 187 5.2. ApproximatemodifiedChapman–Enskogmethod 195 5.3. Kineticcoefficientcalculationwithtakingintoaccountthestatisticalfluctuations 208 Chapter6. SomeApplicationsoftheGeneralizedBoltzmannPhysicalKinetics 215 6.1. InvestigationofthegeneralizedBoltzmannequationforelectronenergydistributioninaconstant electricfieldwithdueregardforinelasticcollisions 215 6.2. Soundpropagationstudiedwiththegeneralizedequationsoffluiddynamics 226 6.3. Shockwavestructureexaminedwiththegeneralizedequationsoffluiddynamics 238 Chapter7. NumericalSimulationofVortexGasFlowUsingtheGeneralizedEuler Equations 241 7.1. Unsteadyflowofacompressiblegasinacavity 241 7.2. Applicationofthegeneralizedhydrodynamicequations:totheinvestigationofgasflowsinchannels withastep 254 7.3. Vortexandturbulentflowofviscousgasinchannelwithflatplate 266 Chapter8. GeneralizedBoltzmannPhysicalKineticsinPhysicsofPlasmaand Liquids 287 8.1. ExtensionofgeneralizedBoltzmannphysicalkineticsforthetransportprocessesdescriptionin plasma 287 8.2. DispersionequationsofplasmaingeneralizedBoltzmanntheory 297 8.3. Generalizeddispersionrelationsforplasma:theoryandexperiment 313 8.4. Tothekineticandhydrodynamictheoryofliquids 323 Appendices A1. DerivationofenergyequationforinvariantEα=mαVα2/2+εα 339 A2. Three-diagonalmethodofGausseliminationtechnicsforthedifferentialthird-orderequation 347 A3. SomeintegralcalculationsinthegeneralizedNavier–Stokesapproximation 352 A4. Three-diagonalmethodofGausseliminationtechniqueforthedifferentialsecond-orderequation 354 A5. Characteristicscalesinplasmaphysics 356 A6. DispersionrelationsinthegeneralizedBoltzmannkinetictheoryneglectingtheintegralcollision term 357 References 361 SubjectIndex 367 Preface Forabout130yearstheBoltzmannequationhasbelongedtothefundamentalequations ofphysics.Thedestinyofthisequationisasdramaticasthatofitsgreatcreator.Even inBoltzmann’sdaystherewasacompleteawarenessthathisequationacquiresafun- damentalimportanceforphysicsandthatits rangeofvaliditystretchesfromtransport processesandhydrodynamicsallthewaytocosmology–thusfullyjustifyingthekeen attentionitattractedanddebatesitprovoked.Bothsidesofthedisputehaveexhausted theirarguments.Thus,thedevelopmentofBoltzmannkinetictheoryhasturnedouttobe typicalofanyrevolutionaryphysicaltheory–fromrejectiontorecognitionandfurther toakindof“canonization”. About twenty years ago it was shown by the author of this book that taking into account the variation of the distribution function over times of the order of the colli- siontimeledtoadditionaltermsintheBoltzmannequation,whichwereproportionalto meantimebetweencollisionsofparticlesandthereforetotheKnudsennumberandvis- cosityinthehydrodynamiclimitofthetheory.Moreover,itturnsoutthattheseterms– whose influence grows with an increase in the Knudsen number – cannot be omitted inthecaseofsmallKnudsennumbersbecausethesetermscontainsmallparametersin frontof senior derivatives.Then these terms should be conservedin the theory in the whole diapasonof evolutionof Knudsennumbers.I have beenworkingin the kinetic theoryformorethan40yearsandthisconclusionwasdramaticfirstandforemostfor myself. Therefore,thecaseinpointisofunprecedentedsituationinphysics,whenthefun- damentalphysicalequationis revised.Duringmystay in Marseille as an InvitedPro- fessor A.J.A. Favre reminded me Henri Poincaré’s phrase after the death of a great Austrianphysicist:“Boltzmannwaswrong,buthismistakeisequaltozero”.It’sapity, butthe situationin the kinetictheoryis moreserious.Obviously,changingthe funda- mentalequationleads–tosomeextent–topossiblechangesoftheknownresultsinthe moderntransporttheoryofphysics.Thisbookreflectsthescaleofthesealterations.It issafetosay–asthemainresultofthegeneralizedBoltzmannkinetictheory–thatthis theoryhasshowedittobeahighlyeffectivetoolforsolvingmanyphysicalproblemsin theareaswheretheclassicaltheoryrunsintodifficulties. TheauthorisdeeplyindebtedtoV.L.GinzburgandF.Uhligfortheirinterestinthis workandinthesubjectingeneral.IamthankfultoV.MikhailovandA.Fedoseyevfor theircooperation. October,2003 vii This page intentionally left blank “AllesVergängliche istnureinGleichniss!” Boltzmann’sepigraph forhis“VorlesungenüberGastheorie” Historical Introduction and the Problem Formulation In1872L.Boltzmann,thenamere28yearsold,publishedhisfamouskineticequation fortheone-particledistributionfunctionf(r,v,t)(Boltzmann,1872).Heexpressedthe equationintheform Df =Jst(f), (I.1) Dt whereJstisacollision(“stoß”)integral,and D ∂ ∂ ∂ = +v· +F· (I.2) Dt ∂t ∂r ∂v isasubstantial(particle)derivative,vandrbeingthevelocityandtheradius-vectorof theparticle,respectively. Eq.(I.1)governsthetransportprocessesinaone-componentgaswhichissufficiently rarefiedsothatonlybinarycollisionsbetweenparticlesareofimportance.Whileweare notconcernedherewiththeexplicitformofthecollisionintegral(whichdeterminesthe changeofthedistributionfunctionf inbinarycollisions),notethatitshouldsatisfythe conservationlaws. For the simplest case of elastic collisions in a one-componentgas, wehave (cid:1) Jstψ dv=0 (i=1,2,3), dv=dv dv dv , (I.3) i x y z whereψ arecollisionalinvariants(ψ =m,ψ =mv,ψ =mv2/2,m isthemassof i 1 2 3 theparticle)relatedtothelawsofconservationofmass,momentum,andenergy. Integralsof the distributionfunction(i.e.,its moments)determinethe macroscopic hydrodynamiccharacteristicsofthesystem,inparticular,thenumberdensityofparti- cles (cid:1) n= f dv (I.4) andthetemperatureT: (cid:1) 3 1 k Tn= m f(v−v )2dv. (I.5) B 0 2 2 1