Table Of ContentATLANTIS STUDIES IN MATHEMATICS FORENGINEERING AND SCIENCE
VOLUME2
SERIES EDITOR: C.K.CHUI
Atlantis Studies in
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AMSTERDAM –PARIS
(cid:2)c ATLANTISPRESS/WORLDSCIENTIFIC
Mean Field Theories and Dual Variation
Takashi SUZUKI
DivisionofMathematicalScience
DepartmentofSystemsInnovation
GraduateSchoolofEngineeringScience
OsakaUniversity
Osaka
Japan
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(cid:2)c 2008ATLANTISPRESS/WORLDSCIENTIFIC
Preface
In general, systems of nonlinearpartial differentialequationsare formulatedand studied
individually according to sepcific physical or biological models of interest. There are,
however,certain classes of Lagrangiansystemsfor which the asymptoticbehaviorof the
non-stationarysolutions,includingtheblowupmechanism,iscontrolledbytheirtotalsets
of stationary solutions. In this book, we will show that these equationshave their origin
inmeanfieldtheoryandthattheassociatedellipticproblemsareprovidedwithmassand
energyquantizationsbecauseoftheirscalingproperties.
Meanfieldapproximationhasbeenadoptedtodescribemacroscopicphenomenafrommi-
croscopicoverviews,withsomesuccess,thoughstillinprogress,inmanyareasofscience,
such as the study of turbulence, gauge field, plasma physics, self-interacting fluids, ki-
netictheory,distributionfunctionmethodinquantumchemistry,tumorgrowthmodelling,
phenomenologyofcriticalphenomena. Inthislastscientific area,phasetransition,phase
separation,andshapememoryalloysareincluded.
However,inspiteofsuchawiderangeofscientificareasthatareconcernedwiththemean
fieldtheory,aunifiedstudyofitsmathematicalstructurehasnotbeendiscusssedexplicitly
intheopenliterature. Thebenefitofthispointofviewonnonlinearproblemsshouldhave
siginificantimpactonfutureresearch,aswillbeseenfromtheunderlyingfeaturesofself-
assembly or bottom-upself-organizationwhich is to be illustrated in a unified way. The
aimofthisbookistoformulatethevariationalandhierarchicalaspectsoftheequationsthat
arise in the mean field theory from macroscopic profiles to microscopic principles, from
dynamicstoequilibrium,andfrombiologicalmodelstomodelsthatarisefromchemistry
andphysics.
Oneofthekeyconceptstobeusedrepeatedlyinourdiscussioninthisbookisthenotionof
duality,whichhasbeentheoriginoftheextensionoffunctionstodistributions273thatpro-
videsausefultoolfortheformulationofweaksolutionsinthetheoryofpartialdifferential
equations. Inaddition,Riesz’representationtheorem100providestheformulationofduals
forsuchimportantfunctionspacesasC(K)(cid:3)∼=M(K)andL1(Ω)(cid:3)∼=L∞(Ω),whereKandΩ
arecompactandopensetsinRn,respectively.Thusifafamilyofcontinuousapproximate
solutionsisnotcompactbutisprovidedwithaprioriestimates,say,intheL∞ orL1 norm,
∞
then the weak solution is realized as an L functionor a measure, respectively, whereby
several fundamental profiles caused by the nonlinearity can be observed (e.g., interface,
shock, localization, concentration, condensate, blowup, self-similar evolution, travelling
andspiralwaves,...).Anotherconceptofduality,namely,thatofHardy-BMO288isreflex-
v
vi Preface
ive. ItisrepresentedbyLlogL(Ω)(cid:3)∼=Exp(Ω)andExp(Ω)(cid:3)∼=LlogL(Ω), whereΩ⊂Rn
isaboundeddomain. TheHardy-BMOdualityisassociatedwiththemeanfieldhierarchy
through the entropy functional and the Gibbs measure. Thus this duality notion can be
applied to the study of phenomena between particles and fields in the microscopic level
as governedby the Boltzmann principle and the zeta-function, respectively. Yet another
dualityconceptthatwillbeusedinthisbookisoperatortheoretical.Forexample,forsome
givenrectangularmatrixAacolumnvectorb,thedualityofthelinearmodelAx=bisthe
transposedoperationthatarisesinthecontextoftheFredholmformulationconcerningthe
existenceanduniquenessofthesolution. Ontheotherhand,thelinearinequalityAx≥b
inconvexanalysishasananalogousdualityformulation.Forexample,thisformulationre-
sultsfromintegratingalinearfunctionalequationwheretheLegendretransformationtakes
ontheroleofthetransposedoperator.30 Thisinvolutiveoperationisoftenreferredtoasthe
Fenchel-Moreauduality,whichresultsin the morecomplicateddualitiesof Kuhn-Tucker
andToland. Here, theTolanddualityisconcernedwith thefunctionalrepresentedbythe
differenceof two convexfunctionals, such as the Helmholtz free energy. Thefield func-
tionalthenisobtainedinitsdualform,andthesetwoprincipalfunctionalsareregardedas
unfoldingsoftheLagrangian.
Thisbookconsistsoftwomainchapters. Thefirstchapterisdevotedtothestudyofvaria-
tionalstructures,wherethequantizedblowupmechanismobservedinthechemotaxissys-
tem304 is first described. Within this section, the free energytransmission, which results
intheformationofself-assembly,isemphasizedwhichmeansthatthesourceoftheemer-
genceis the wedge ofblowupenvelope,while entropyand massare exchangedto create
a clean self with the quantized mass. The leading principle derived from this study can
be summarized as follows: dual variation sealed in the mean field hierarchyand scaling
that reveals this hierarchy. In fact, the phenomenonof quantizationis sealed in the total
set of stationarysolutions, referredto as the nonlinearspectralmechanics. The structure
ofdualvariationcontrols,notonlythelocaldynamicsaroundthestationarystate,butalso
theformationofself-assemblyrealizedasa globaldynamicsincludingtheblowupofthe
solution. Thisdualityisassociatedwiththeconvexityofthefunctional;andthus,ourde-
scriptioninvolvesconvexanalysisas mentionedabove. We will examinethe phenomena
and logistics of both variational and dynamical structures of several problems including
phase transition, phase separation, hysteresis in shape memory alloys which review the
theoryofnon-equilibriumthermodynamics.
Chapter 2 is devoted to the discussion of the method of scaling for revealing the mathe-
maticalprincipleshiddeninthemeanfieldhierarchysubjecttothesecondlawofthermo-
dynamics. Moreprecisely,wewilldescribethequantizedblowupmechanismobservedin
self-gravitating fluid, turbulence, and self-dual gauge field. In particular, blowup anlay-
sis,oneofthemostimportantapplicationsofthemethodofscaling,isusedtoclarifythis
mechanism. This analysis consists of four ingredients, namely, scaling invarianceof the
problem,classificationofthesolutionofthelimitproblemdefinedonthewholespaceand
time, control of the rescaled solution at infinity, and hierarchicalarguments. Meanwhile
we will also describe the method of statistical mechanics for the study of macro-scopic
Preface vii
phenomenacausedbythemicro-scopicprinciples. Inparticular,wewillre-formulatethe
simplified system of chemotaxisas a fundamentalequationof the materialtransport, the
Smoluchowski-Poissonequation.
Unfortunately, a complete overview of the topics mentioned above is beyond the scope
of this book, and the list of referencesis far from being sufficient. Furthermore, several
importantrelatedtopicsaremissing,amongwhicharethecomplexGinzburg-Landaufree
energy associated with super-conductivity in low temperature,22,227 Arnold’s variational
principlefordescribinigthe MHD equilibrium,11 the densityfunctiontheoryofquantum
chemistry,43kineticequationsinfluiddynamics,29andthedualitybetweenAleksandrov’s
probleminintegralgeometryandtheoptimalmasstransporttheory.238 Therearealsosev-
eralstationaryproblemsinengineeringassociatedwithduality113whichmaybeextended
tothenon-stationaryproblemsoftheirown.
We take this opportunity to clarify several terminologies used in this book that are the
concernofthermalphenomenaassociatedwith dissipation. First, wefollowthe classical
conceptsofequilibriumthermodynamicsandclassifythermalsystemsintoisolated,closed,
and open systems. This classification is also associated with the theory of equilibrium
statistical mechanics. More precisely, these systems are provided with the microscopic
structuresofmicro-canonical,canonical,andgrand-canonicalensembles,respectively. A
closedsystemhereindicatesthelackoftransportofmaterialsbetweentheoutersystems,
wherebythetransportofheatandthatoftheenergyarepermitted. Intheopensystem,on
theotherhand,thetransportofmaterialbetweentheoutersystemsisalsopermitted.Next,
inviewofthetheoryofnon-equilibriumthermodynamics,weaddonemoreaspecttothe
openness,thatisthedissipationofentropyto theoutersystem.224 Underthisagreement,
closednessisre-formulatedasasystemprovidedwithdecreasingtotalfreeenergyorthat
providedwith increasing total entropy. Thus we have two kinds of closedness: physical
closednessintermsofkineticand/ormaterial,andthermodynamicalclosednessdescribed
bytwotypicalmodelsdiscussedindetailsinthisbook.Freeenergytransmissionoccursin
thelattercaseandcouldbetheoriginof”self-assembly”.
Inthisconnection,wenotethatthedissipativesystemisdefinedbythepresenceofanat-
tractorinthetheoryofdynamicalsystems. Gradientsystemwithcompactsemi-orbitsisa
typicaldissipativesysteminthiscase.128 We believe,however,thatthisdefinitionofdis-
sipativesystemdoesnotdescribewhatwereobservedby.224 Moreprecisely,decreaseof
thefreeenergyorincreaseoftheentropymeansthermodynamicalclosedness. Itisformu-
lated bythegradientsystem in the sametwo modelsasmentionedabove,and, therefore,
isneverprovidedwiththedissipationinthesenseofnon-equilibriumthermodynamics224
in spite of the fact that these two models are typical dissipative systems in the theory of
dynamical systems. To avoid this confusion, the above mentioned terminology of dissi-
pativesystem in thetheoryofdynamicalsystems128 isnotusedin thisbook. Actually, a
recentparadigmasserts two aspects of self-organization,far-from-equilibrium(top-down
self-organization)and self-assembly (bottom -upself-organization),emphasizingthe role
oftheirhierarchicaldevelopments,withtheHopfbifurcationcastingthethreshold.345 Dis-
sipationoccursinastatefar-from-equilibrium,whichresultsinaspiralwave,atravelling
viii Preface
wave,aperiodicstructure,asef-similardevelopment,andsoforth;whileself-assembly,we
believe,isformednear-from-equilibriumofthermodynamicallyclosedsysteminducedby
thestationarystates,developingacondensate,acollapse,ablowup,aspike,andsoforth.
Thisbookisconcernedwiththeformationofself-assembly.Itsprofileisprovidedwiththe
”tripleseal,” ofwhichbasic conceptshavealreadybeendescribed. First, severalfeatures
ofself-assemblythatisoneaspectoftheself-organizationaresealedinthermodynamically
closedsystems.Secondly,thedynamicsoftheclosedsystemissealedinthetotalsetofsta-
tionarystates. Finally,thestationarystatesthemselvesaresealedinthe(skew-)Lagrangian,
providedwith the structure of dual variation. These features of the mean field hierarchy
are certainly revealed by the method of scaling derived from the microscopic principle.
Insummary,formationofself-assemblyarisesinthermodynamicallyclosedsystems,and
hencethisprocessissubjecttothecalculusofvariation.Theclosedsystemfollowsthemi-
croscopicprinciple.Thisprinciplecontrolsthemeanfieldhierarchytotally,and,therefore,
fundamentalprofilesof the macroscopicmean field equation are revealedby the method
ofscaling. WewouldliketothankProfessorTomohikoYamaguchiforseveralstimulative
discussionsonnon-equilibriumthermodynamics. ThanksarealsoduetoProfessorsBiotr
Biler,FumioKikuchi,FutoshiTakahashi,andGershonWolanskyforcarefulreadingover
theprimaryversionofthemanuscript.
TakashiSuzuki
Osaka,Japan
June,30,2008
Contents
Preface v
Chapter1. Duality-SealedVariation 1
1.1 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 ThermalTransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 CollapseFormation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 MeanFieldHierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 StationaryState . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.5 LocalizationandSymmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.6 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.7 MassQuantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.8 BlowupRate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1.9 BlowupinInfiniteTime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.10 TimeRelaxization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.1.11 GeometricFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.1.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2 TolandDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2.1 ExponentialNonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2.2 FullSystemofChemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.2.3 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.2.4 VariationalEquivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.2.5 SpectralEquivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.2.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.2.7 GradientSystemswithDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.2.8 EntropyFunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.2.9 ZygmundSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.3.1 Non-ConvexEvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.3.2 GradientandSkew-GradientSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.3.3 Skew-GradientSystemswithDuality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.3.4 Semi-Unfolding-Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1.3.5 Kuhn-TuckerDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.4.1 StefanProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
1.4.2 ThermalEquilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
1.4.3 Penrose-FifeTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
1.4.4 Non-Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
1.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
1.5 PhaseFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
1.5.1 Fix-CaginalpEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
ix
