Microscopic Dynamics of Plasmas and Chaos

327 Pages·2003·2.864 MB·English

by  Yves Elskens, Dominique Escande

Microscopic Dynamics of Plasmas and Chaos
Series in Plasma Physics
SeriesEditors:
SteveCowley,ImperialCollege,UK
PeterStott,CEACadarache,France
HansWilhelmsson,ChalmersUniversityofTechnology,Sweden
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Series in Plasma Physics
Microscopic Dynamics of Plasmas
and Chaos
Yves Elskens
CNRS—Universite´ de Provence (Marseilles)
Dominique Escande
CNRS—Universite´ de Provence (Marseilles)
Institute of Physics Publishing
Bristol and Philadelphia
(cid:1)c IOPPublishingLtd2003
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Iknowthetendencyofthehumanmindtodoanythingratherthanthink.None
ofusexpecttosucceedwithoutlabour,andweallknowthattolearnanyscience
requires mental labour, and I am sure we would all give a great deal of mental
labour to get up our subjects. But mental labour is not thought, and those who
havewithgreatlabouracquiredthehabitofapplication,oftenfinditmucheasier
togetupaformulathantomasteraprinciple.Ishallendeavourtoshowyouhere,
whatyouwillfindtobethecaseafterwards,thatprinciplesarefertileinresults,
butthemereresultsarebarren,andthatthemanwhohasgotupaformulaisat
the mercy of his memory, while the man who has thought out a principle may
keephismindclearofformulæ,knowingthathecouldmakeanynumberofthem
whenrequired.
(JamesClerkMaxwell,
InaugurallectureatKing’sCollege,
London,October1860)
Permissions
Figuresreprintedfrom
• Phys. Plasmas 4 Be´nisti and Escande (cid:2)c1997 with permission from the
AmericanInstituteofPhysics
• Phys.Lett.A284Doveiletal(cid:2)c2001withpermissionfromElsevierScience
• Physica D 62 Elskens and Escande (cid:2)c1993 with permission from Elsevier
Science
• Nonlinearity 4 Elskens and Escande (cid:2)c1991 with permission from the
InstituteofPhysics
• Phys. Rev. E 64 Firpo et al (cid:2)c2001 with permission from the American
PhysicalSociety
• Phys. Rev. Lett. 84 Firpo and Elskens (cid:2)c2000 with permission from the
AmericanPhysicalSociety
• Transport,ChaosandPlasmaPhysicsvol2,Guyomarc’hetal(cid:2)c1996with
permissionfromWorldScientificPublishingCo
vi
Contents
Permissions vi
Preface xiii
Difficultiesinthetraditionalapproach xiii
Benefitsofmoderntools xv
Structureofthebook xvi
Prerequisitesandremarks xviii
1 Basicphysicalsetting 1
1.1 TheoriginalN-bodysystem 1
1.1.1 Physicalcontext 1
1.1.2 Theplasmamodel 2
1.2 Wave–particleresonance:aparadigmofclassicalmechanics 5
2 FromN-bodydynamicstowave–particleinteraction 10
2.1 Intuitivederivationoftheself-consistentHamiltonian 11
2.2 Langmuirwaveswithoutresonantparticles 13
2.2.1 Decompositionofthefieldandparticlemotion—relevant
smallparameters 13
2.2.2 Collectivedynamics 15
2.2.3 Bohm–Grossmodes 17
2.3 Coupledmotionofquasi-resonantparticleswithBohm–Gross
modes 20
2.4 Lagrangianformulation* 24
2.5 Referencestatesoftheplasma* 28
2.5.1 Non-existenceofzero-fieldstates 28
2.5.2 Thermaldistributionoftheelectricfield 29
2.6 Physicalscalingsanderrorestimates* 31
2.7 FinalformoftheHamiltonian 37
2.8 Historicalbackgroundandnotes 38
viii Contents
3 Dynamicsofthesmall-amplitudewave–particlesystem 40
3.1 Referencestatewithavanishingelectricfield 41
3.2 Smallperturbationtothereferencestate 42
3.3 Ballisticsolutions 46
3.4 Wavelikesolutions 47
3.5 Initialvalueproblem 49
3.6 Dispersionrelationforwavelikemodes 51
3.7 Physicalinterpretation:coldbeams 52
3.7.1 Caseofasinglebeam 52
3.7.2 Caseoftwobeams 55
3.8 Manycoldbeamsorawarmbeam 57
3.8.1 Landauunstablemode 57
3.8.2 Eigenmodesandinitialvalueproblem 59
3.8.3 VanKampenmodes* 61
3.8.4 Relationwithballisticeigenmodes 68
3.8.5 Transitionfromcoldtowarmbeams 69
3.9 Synchronizationofparticleswithawave 69
3.9.1 Synchronizationofparticleswithawavelikemode 69
3.9.2 SynchronizationofparticlesduringLandaudamping* 71
3.9.3 Fateofparticlesinthepresenceofmanyincoherentmodes 72
3.10 Historicalbackground 72
4 Statisticaldescriptionofthesmall-amplitudewave–particledynamics 74
4.1 Approachusingperturbationexpansion 75
4.1.1 Second-orderperturbationanalysis 75
4.1.2 Evolutionofwaves 77
4.1.3 Evolutionofparticles 81
4.1.4 Fokker–Planckequationfortheparticles 82
4.2 ApproachusingFloquetequation* 85
4.3 Linkwithtraditionaldescriptions 90
4.3.1 LandaueffectfromaVlasovianpointofview 90
4.3.2 Spontaneousemission 92
5 Hamiltonianchaos 94
5.1 GeometricaltoolsforHamiltonianchaos 96
5.1.1 Poincare´surfaceofsection 96
5.1.2 Conservationofareas,symplecticdynamicsandflux* 99
5.1.3 Action-anglevariables 101
5.2 Motionofoneparticleinthepresenceoftwowaves 103
5.2.1 Smallresonanceoverlapandcantori 104
5.2.2 Moderateresonanceoverlapandstochasticlayers 106
5.2.3 Physicalsummary 109
5.3 Constructionoforbits 111
5.3.1 Originofhigher-orderresonances 111
5.3.2 Poincare´andKAMtheorems* 114
Contents ix
5.3.3 Higher-orderresonancesfromaction-anglevariables 115
5.4 RenormalizationforKAMtori 117
5.4.1 Simpleapproachtorenormalization 117
5.4.2 Moreexplicitderivation* 119
5.4.3 Studyoftherenormalizationmapping 123
5.4.4 Thresholdsandexponents 125
5.5 Orderinchaos 129
5.5.1 WigglingarmsoftheX-point 129
5.5.2 Largeresonanceoverlap:numericalresults 133
5.5.3 Largeresonanceoverlap:analyticalresults* 136
5.6 Historicalbackgroundandfurthercomments 139
6 Diffusion:thecaseofthenon-self-consistentdynamics 141
6.1 ModelHamiltonian 142
6.2 Diffusionasanumericalfact 142
6.3 Conceptofresonancebox 148
6.3.1 Heuristicanalyticalapproach 149
6.3.2 Numericalcheckoftheconcept 153
6.3.3 Rigorousapproach* 154
6.4 Scalingpropertiesoffinite-timedynamics 156
6.5 Originoftheforcedecorrelation 158
6.5.1 Localityandresonanceboxesforrandomphases 158
6.5.2 Correlatedphases 160
6.5.3 Randominitialpositions 161
6.6 Originofthechaoticdiffusion 161
6.7 LocalityanddiffusionformoregeneralHamiltonians 163
6.8 Initialquasilineardiffusion 166
6.8.1 Non-chaoticquasilineardiffusion 167
6.8.2 Initialquasilineardiffusionuptochaos 169
6.9 Diffusioncoefficientinthechaoticregime 174
6.10 Dragcoefficient 177
6.11 Fokker–Planckequation 181
6.12 Historicalbackground 182
7 Self-consistentdynamicsinthediffusiveregime 183
7.1 Quasilineardiffusioncoefficient 185
7.2 Simplederivationofthequasilinearequations 187
7.3 Evolutionofwaves* 189
7.4 Dragonparticles* 195
7.5 Jointevolutionofparticlesandwaves 197
7.6 Saturationoftheweak-warm-beaminstability 198
7.7 Historicalbackgroundandfurthercomments 203