Pattern Formation and Dynamics in Nonequilibrium Systems

553 Pages·2009·7.102 MB·English

by  Cross M., Greenside H.

PATTERN FORMATION AND DYNAMICS
IN NONEQUILIBRIUM SYSTEMS
Manyexcitingfrontiersofscienceandengineeringrequireunderstandingofthespa-
tiotemporalpropertiesofsustainednonequilibriumsystemssuchasfluids,plasmas,
reactinganddiffusingchemicals,crystalssolidifyingfromamelt,heartmuscle,and
networksofexcitableneuronsinbrains.
Thisintroductorytextbookforgraduatestudentsinbiology,chemistry,engineer-
ing, mathematics, and physics provides a systematic account of the basic science
commontothesediverseareas.Thisbookprovidesacarefulpedagogicalmotivation
ofkeyconcepts,discusseswhydiversenonequilibriumsystemsoftenshowsimilar
patternsanddynamics,andgivesabalanceddiscussionoftheroleofexperiments,
simulation,andanalytics.Itcontainsnumerousillustrativeworkedexamples,and
over150exercises.
This book will also interest scientists who want to learn about the experi-
ments,simulations,andtheorythatexplainhowcomplexpatternsforminsustained
nonequilibriumsystems.
Michael Cross isaProfessorofTheoreticalPhysicsattheCaliforniaInsti-
tuteofTechnology,USA.Hisresearchinterestsareinnonequilibriumandnonlinear
physics including pattern formation, chaos theory, nanomechanical systems, and
condensedmatterphysics,particularlythetheoryofliquidandsolidhelium.
Henry Greenside is a Professor in the Department of Physics at Duke
University,USA.Hehascarriedoutresearchincondensedmatterphysics,plasma
physics,nonequilibriumpatternformation,andtheoreticalneurobiology.Heisalso
involved with outreach programs to stimulate interest in science and physics at
juniorhighschoolandhighschoollevels.
“This book by Cross and Greenside presents a comprehensive introduction to an
importantareaofnaturalscience,andassemblesinonevolumetheessentialcon-
ceptual,theoretical,andexperimentaltoolsaseriousstudentwillneedtoobtaina
modern understanding of pattern formation outside of equilibrium.The masterful
50-pageIntroductionlaysouttheessentialquestionsandprovidesmotivationtothe
readertoexplorethesubsequentchapters,beginningwithsimpleideasandgrowing
progressivelyinmathematicalsophisticationandphysicaldepth.Carefulattention
is paid to the relationship between the theoretical methods and controlled labora-
tory experiments or numerical simulations. I can highly recommend this book to
anystudentorresearcherinterestedinadeepenedunderstandingofnonequilibrium
spatiotemporalpatterns.’’
PierreHohenberg,NewYorkUniversity
“Thisbookgivesanexcellentdidacticintroductiontopatternformationinspatially
extended systems. It can serve both as the basis for an advanced undergraduate
or graduate course as well as a reference. It is one of those books that will never
outliveitsusefulness.Itisamustforanyoneinterestedinnonlinear,nonequilibrium
physics.’’
EberhardBodenschatz,MPIforDynamicsandSelf-Organization,
UniversityofGoettingen,CornellUniversity
“This book fills a long-standing need, and is certain to be an instant classic. The
physicsofpatternformingsystemsisdiversebutthetheoreticalcoreofthesubject,
along with many of the most important applications, can be learned from this
splendid book. It is bound to be a key text for courses, as well as a much cited
reference.’’
StephenMorris,UniversityofToronto
PATTERN FORMATION
AND DYNAMICS IN
NONEQUILIBRIUM SYSTEMS
MICHAEL CROSS
CaliforniaInstituteofTechnology
HENRY GREENSIDE
DukeUniversity
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Contents
Preface page xiii
1 Introduction 1
1.1 Thebigpicture:whyistheUniversenotboring? 2
1.2 Convection:afirstexampleofanonequilibriumsystem 3
1.3 Examplesofnonequilibriumpatternsanddynamics 10
1.3.1 Naturalpatterns 10
1.3.2 Preparedpatterns 20
1.3.3 Whataretheinterestingquestions? 35
1.4 Newfeaturesofpattern-formingsystems 38
1.4.1 Conceptualdifferences 38
1.4.2 Newproperties 43
1.5 Astrategyforstudyingpattern-formingnonequilibrium
systems 44
1.6 Nonequilibriumsystemsnotdiscussedinthisbook 48
1.7 Conclusion 49
1.8 Furtherreading 50
2 Linearinstability:basics 56
2.1 Conceptualframeworkforalinearstabilityanalysis 57
2.2 Linearstabilityanalysisofapattern-formingsystem 63
2.2.1 One-dimensionalSwift–Hohenbergequation 63
2.2.2 Linearstabilityanalysis 64
2.2.3 Growthratesandinstabilitydiagram 67
2.3 Keystepsofalinearstabilityanalysis 69
2.4 Experimentalinvestigationsoflinearstability 70
2.4.1 Generalremarks 70
2.4.2 Taylor–Couetteinstability 74
vii
viii Contents
2.5 Classificationforlinearinstabilitiesofauniformstate 75
2.5.1 Type-Iinstability 77
2.5.2 Type-IIinstability 79
2.5.3 Type-IIIinstability 80
2.6 Roleofsymmetryinalinearstabilityanalysis 81
2.6.1 Rotationallyinvariantsystems 82
2.6.2 Uniaxialsystems 84
2.6.3 Anisotropicsystems 86
2.6.4 Formaldiscussion 86
2.7 Conclusions 88
2.8 Furtherreading 88
3 Linearinstability:applications 96
3.1 Turinginstability 96
3.1.1 Reaction–diffusionequations 97
3.1.2 Linearstabilityanalysis 99
3.1.3 Oscillatoryinstability 108
3.2 Realisticchemicalsystems 109
3.2.1 Experimentalapparatus 109
3.2.2 Evolutionequations 110
3.2.3 Experimentalresults 116
3.3 Conclusions 119
3.4 Furtherreading 120
4 Nonlinearstates 126
4.1 Nonlinearsaturation 129
4.1.1 Complexamplitude 130
4.1.2 Bifurcationtheory 134
4.1.3 NonlinearstripestateoftheSwift–Hohenbergequation 137
4.2 Stabilityballoons 139
4.2.1 Generaldiscussion 139
4.2.2 BusseballoonforRayleigh–Bénardconvection 147
4.3 Two-dimensionallatticestates 152
4.4 Non-idealstates 158
4.4.1 Realisticpatterns 158
4.4.2 Topologicaldefects 160
4.4.3 Dynamicsofdefects 164
4.5 Conclusions 165
4.6 Furtherreading 166
Contents ix
5 Models 173
5.1 Swift–Hohenbergmodel 175
5.1.1 Heuristicderivation 176
5.1.2 Properties 179
5.1.3 Numericalsimulations 183
5.1.4 Comparisonwithexperimentalsystems 185
5.2 GeneralizedSwift–Hohenbergmodels 187
5.2.1 Non-symmetricmodel 187
5.2.2 Nonpotentialmodels 188
5.2.3 Modelswithmeanflow 188
5.2.4 Modelforrotatingconvection 190
5.2.5 Modelforquasicrystallinepatterns 192
5.3 Order-parameterequations 192
5.4 ComplexGinzburg–Landauequation 196
5.5 Kuramoto–Sivashinskyequation 197
5.6 Reaction–diffusionmodels 199
5.7 Modelsthatarediscreteinspace,time,orvalue 201
5.8 Conclusions 201
5.9 Furtherreading 202
6 One-dimensionalamplitudeequation 208
6.1 Originandmeaningoftheamplitude 211
6.2 Derivationoftheamplitudeequation 214
6.2.1 Phenomenologicalderivation 214
6.2.2 Deductionoftheamplitude-equationparameters 217
6.2.3 Methodofmultiplescales 218
6.2.4 Boundaryconditionsfortheamplitudeequation 219
6.3 Propertiesoftheamplitudeequation 221
6.3.1 Universalityandscales 221
6.3.2 Potentialdynamics 224
6.4 Applicationsoftheamplitudeequation 226
6.4.1 Lateralboundaries 226
6.4.2 Eckhausinstability 230
6.4.3 Phasedynamics 234
6.5 Limitationsoftheamplitude-equationformalism 237
6.6 Conclusions 238
6.7 Furtherreading 239
7 Amplitudeequationsfortwo-dimensionalpatterns 244
7.1 Stripesinrotationallyinvariantsystems 246
7.1.1 Amplitudeequation 246
7.1.2 Boundaryconditions 248