Table Of ContentThe Vlasov Equation 1
Series Editors
Pierre-Noël Favennec and Frédérique de Fornel
The Vlasov Equation 1
History and General Properties
Pierre Bertrand
Daniele Del Sarto
Alain Ghizzo
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter1.IntroductiontoaUniversalModel: theVlasov
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.Ahistoricalpointofview . . . . . . . . . . . . . . . . . . . . . . 1
1.2.Individualandcollectiveeffectsinplasmas . . . . . . . . . . . . 5
1.3.Graininessparameter . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.ThecollectivedescriptionofaCoulombgas: anintuitive
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.FromN-bodytoVlasov . . . . . . . . . . . . . . . . . . . . . . 12
1.6.Thegraininessparameterand1D,2Dor3Dmodels . . . . . . . 16
1.7.TheVlasovequationatthemicroscopicfluctuationslevel . . . 19
1.8.TheWignerequation(Vlasovequationforquantumsystems) . 21
1.9.TherelativisticVlasov–Maxwellmodel . . . . . . . . . . . . . . 26
1.10.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter2.AParadigmforaCollectiveDescriptionof
aPlasma: the1DVlasov–PoissonEquations . . . . . . . . . . . 31
2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.ThelinearLandauproblem . . . . . . . . . . . . . . . . . . . . . 33
2.2.1.TheMaxwelliancase . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2.Landaupolesandothers . . . . . . . . . . . . . . . . . . . . 36
2.2.3.Unstableplasma: two-streaminstability . . . . . . . . . . . 38
2.3.The1Dcoldplasmamodel: nonlinearoscillations . . . . . . . . 39
2.3.1.Hydrodynamicdescription . . . . . . . . . . . . . . . . . . . 39
2.3.2.LagrangianformulationthroughtheVonMises
transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vi TheVlasovEquation1
2.3.3.Thewave-breakingphenomenon. . . . . . . . . . . . . . . . 42
2.4.Thewaterbagmodel . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1.Basicequations . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2.Linearizedtheory . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.3.Waterbaghydrodynamicdescription . . . . . . . . . . . . . 48
2.5.Connectionbetweenthehydrodynamic,waterbagand
Vlasovmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.1.AVlasovhydrodynamicdescription. . . . . . . . . . . . . . 50
2.5.2.VlasovnumericalsimulationsofPn−3 . . . . . . . . . . . . 52
2.5.3.ThefundamentalcontributionofpolesbesidesLandau . . . 56
2.6.Themultiplewaterbagmodel . . . . . . . . . . . . . . . . . . . 58
2.6.1.Amultifluiddescription. . . . . . . . . . . . . . . . . . . . . 59
2.6.2.Linearizedanalysis . . . . . . . . . . . . . . . . . . . . . . . 63
2.7.Furtherremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.8.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter3.ElectromagneticFieldsinVlasovPlasmas:
GeneralApproachtoSmallAmplitudePerturbations . . . . . . 75
3.1.Introductionandoverviewofthechapter . . . . . . . . . . . . . 75
3.2.LinearanalysisoftheVlasov–Maxwellsystem: general
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.1.Dispersionrelationandresponsematrix . . . . . . . . . . . 81
3.2.2.Thechoiceofthebasisfortheresponsetensor . . . . . . . . 83
3.2.3.Aboutthenumberof“waves”inplasmas . . . . . . . . . . . 89
3.2.4.Realorcomplexvaluesofkandω: steadystateandinitial
valueproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.Polynomialapproximationsofthedispersionrelation:whyand
howtousethem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.1.Truncated-Vlasovandfluid–plasmadescriptionsforthe
linearanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3.2.Wavedispersionandresonancesallowedbyinclusionof
high-ordermomentsinfluidmodels . . . . . . . . . . . . . . . . . 99
3.3.3.Anexample: fluidmomentsandFinite–Larmor–Radius
effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3.4.Keypointsaboutapproximatednormalmodeanalysis . . . 108
3.4. Vlasov plasmas as collisionless conductors with polarization
andfiniteconductivity: meaningofplasma’s“dielectrictensor” . . 109
3.4.1.Polarizationchargesandwaveequationindielectric
materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.2.The“equivalent”dielectrictensoranditscomplex
components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Contents vii
3.4.3.Temporalandspatialdispersioninplasmas . . . . . . . . . . 120
3.4.4.Conductivityandcollisionalresistivityin
Vlasovplasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5. Symmetry properties of the complex components of the
equivalentdielectrictensorandenergyconservation . . . . . . . . . 126
3.5.1.Onsager’srelations . . . . . . . . . . . . . . . . . . . . . . . 126
3.5.2.Poynting’stheorem . . . . . . . . . . . . . . . . . . . . . . . 129
3.5.3.Symmetryofthecoefficientsoftheequivalentdielectric
tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.5.4.MoreaboutOnsager’srelationsforwavedispersion . . . . . 134
3.5.5.Energydissipationversusrealandimaginarypartsofσ
ij
and(cid:4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
ij
3.6.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Chapter4.ElectromagneticFieldsinVlasovPlasmas:
CharacterizationofLinearModes . . . . . . . . . . . . . . . . . . 147
4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.2.Characterizationofelectromagneticwavesandof
wave-packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.1.Polarizationofelectromagneticwavesinplasmas . . . . . . 153
4.2.2.Phasevelocity,groupvelocityandrefractiveindex . . . . . 156
4.2.3.Exampleofpropagationinunmagnetizedplasmas:
underdenseandoverdenseregimes . . . . . . . . . . . . . . . . . . 161
4.2.4.Exampleofpropagationinmagnetizedplasmas:
ion-cyclotronresonancesandFaraday’srotationeffect . . . . . . . 166
4.2.5.Wave–particleresonances,Landaudampingandwave
absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2.6.Resonanceandcut-offconditionsontherefractiveindex . . 176
4.2.7.Graphicalrepresentationsofthedispersionrelation . . . . . 178
4.3.InstabilitiesinVlasovplasmas: someterminologyandgeneral
features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.3.1.Linearinstabilities . . . . . . . . . . . . . . . . . . . . . . . . 184
4.3.2.Absoluteandconvectiveinstabilitiesandsomeother
classificationcriteria . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.4. On some complementary interpretations of the collisionless
dampingmechanisminVlasovplasmas . . . . . . . . . . . . . . . . 198
4.4.1.LandaudampingasaninverseVavilov–Cherenkov
radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.4.2.LandaudampinginN-body“exact”models . . . . . . . . . 203
viii TheVlasovEquation1
4.4.3.Somefinalremarksaboutinterpretativeissuesof
collisionlessdampinginVlasovmeanfieldtheory . . . . . . . . . 206
4.5.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Chapter5.NonlinearPropertiesofElectrostaticVlasov
Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.1.TheVlasov–Poissonsystem . . . . . . . . . . . . . . . . . . . . 215
5.2.InvariantsoftheVlasov–Poissonmodel. . . . . . . . . . . . . . 216
5.3.Stationarysolutions: Bernstein–Greene–Kruskalequilibria . . . 217
5.4.SomemathematicalpropertiesoftheVlasovequation. . . . . . 220
5.5.TheBernstein–Greene–Kruskalsolutions . . . . . . . . . . . . . 229
5.5.1.Thecaseof(electrostatic)two-streaminstability. . . . . . . 230
5.5.2.ChainofBGKequilibria . . . . . . . . . . . . . . . . . . . . 235
5.5.3.StabilityoftheperiodicBGKsteadystates . . . . . . . . . . 236
5.6.TravelingwavesofBGK-typesolutions . . . . . . . . . . . . . . 242
5.7.Roleofminoritypopulationoftrappedparticles . . . . . . . . . 245
5.7.1.NonlinearLandaudampingandtheemergenceofthe
nonlinearLangmuir-typewave . . . . . . . . . . . . . . . . . . . . 247
5.7.2.ElectronacousticwaveinthenonlinearLandaudamping
regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.7.3.Kineticelectrostaticelectronnonlinearwaves . . . . . . . . 260
5.7.4.EmergentresonanceforKEENwaves . . . . . . . . . . . . . 268
5.8.NatureofKEENwavesandNMI . . . . . . . . . . . . . . . . . 270
5.8.1.Adiabaticmodelforasinglelinearwave:the(electrostatic)
trappedelectronmodemodel . . . . . . . . . . . . . . . . . . . . . 270
5.8.2.TheDodinandFischmodelconnectedtotheemergenceof
KEENwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
5.9.Electronholeandplasmawaveinteraction . . . . . . . . . . . . 281
5.10.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Preface
Plasmaphysicsisaratheryoungbranchofmodernscience.Thereasonis
thatplasmasarenotnaturallypresentonEarth,exceptduringathunderstorm.
But around 1920–1930, the pioneering works of Irving Langmuir and Lewis
Tonks on electrical discharges pointed to the peculiar properties of a “new
medium”: theplasmastate.Duringthissameperiod,someoftheseproperties
were being indirectly and independently observed in the measurements of
radio-wave transmission through that part of the atmosphere which at that
time was being discovered: the ionosphere, which represents the “gas” in a
natural,persistentplasmastatethatisclosesttoEarth’ssurface.
Inaneutralgas,themainphysicalpropertiescanbeexplainedintermsof
a statistical description of the mutual interactions of a large number of
particles (usually neutrally charged molecules), where each particle interacts
(i.e.“collides”)withjustaveryfewotherparticlesatatime.Onthecontrary,
inaplasma,thepresenceoffreechargedparticles(ionsandelectrons)points
to a different and new physical behavior, since a single charge can interact
withahugenumberofotherchargedparticlesatthesametime.
These collective properties can be explained as due to the long-range
interaction of Coulomb forces, which decays as r−2, while the number of
particlesatadistancerfromatestparticlegrowsasr+2.Thisnumbercanbe
larger than 108 in astrophysical plasmas (sun, stars, interstellar medium etc.)
as well as in laboratory plasmas for thermonuclear fusion experiments (by
magnetic or inertial confinement). In the limit in which this number can be
considered as infinite in a mathematical sense, a master equation can be
derived which makes it possible to describe the collective behavior of
plasmas. This is the well-known (at least by plasma physicists!) Vlasov
equation,introducedbyA.Vlasovin1945andL.D.Landauin1946.
x TheVlasovEquation1
However, the modeling of a Vlasov plasma is not that easy, especially
because of the intrinsic nonlinearities of the equations involved. As this
represents an active field of research, approaching it by relying on modern
scientific literature can be a quite challenging task for a newcomer, despite
several excellent books that already exists about the physics of Vlasov
plasmas. Most books on this subject tend to focus either on the physics of
“warm/diluted plasmas”, to which Vlasov formalism applies, or to more
strictlymathematicalpropertiesoftheVlasovequation,relatedtoissuesmore
relevanttoappliedmathematics(convergence,integrability,etc.).Thepresent
book aims to privilege a discussion of the general properties of the Vlasov
equation and of its applications, which are presented to the reader from a
historicaland–hopefully–pedagogicalpointofview.
Althoughplasmaphysicsitselfisaratheryoungdiscipline,beinglessthan
100 years old, one of us (Pierre Bertrand) has been working on this subject
formorethan50years.Hehasbeenamodestactor,buthehasbeeninvolved
during more than half of the “history” of plasma physics. Thus, it seemed to
us that introducing some of the historical works which were carried out
during all these years (a task which is mainly accomplished in the first two
chapters of the present book) would help the understanding of more current
works, whose discussion is developed in the following chapters: notably, the
implicationsoftheVlasovformalismfordescribingthepropagationofsmall
amplitude electromagnetic waves in a plasma (Chapters 3 and 4), and the
nonlinearbehaviorofthelatter,intheelectrostaticVlasovlimit(Chapter5).
This book is addressed to students in plasma physics and to young
researchers, as well as to whomever wants to get a good understanding of a
Vlasovplasma,inordertobeabletograsptheessenceofitsmainproperties.
PierreBERTRAND
DanieleDELSARTO
AlainGHIZZO
June2019
1
Introduction to a Universal
Model: the Vlasov Equation
1.1. Ahistoricalpointofview
Halfway between the N body model and the usual hydrodynamics, the
Vlasovequation,orbettertheVlasovmodel,describesdifferentmediagoing
from nuclear matter to the expanding universe (via semiconductors, plasmas
andstellardynamicsproblemsandtheintroductionofaquantumcounterpart,
theso-calledWignerequation).Theaimofthisbookistoprovidethereader
with a good knowledge of the Vlasov model, which offers a specific
mathematicaldescriptionfordifferentpartsofphysicsorastrophysics.
Figure1.1. SirJamesHopwoodJeans(1877–1946)
The Vlasov Equation 1: History and General Properties,
First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo.
© ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.
