Generalized Boltzmann Physical Kinetics

377 Pages·2004·2.187 MB·English

by  Boris V. Alexeev

Generalized Boltzmann
Physical Kinetics
BORIS V. ALEXEEV
ELSEVIER
Generalized Boltzmann Physical Kinetics
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Generalized Boltzmann Physical Kinetics
BORIS V. ALEXEEV
MoscowFineChemical TechnologyInstitute
Moscow119571,Russia
2004
Amsterdam–Boston–Heidelberg–London–NewYork–Oxford
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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
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“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