Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics

476 Pages·2009·6.241 MB·English

by  Professor Errico Presutti (auth.)

Theoretical and Mathematical Physics
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EditorialBoard
W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Germany
J.-P.Eckmann,DepartmentofTheoreticalPhysics,UniversityofGeneva,Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA,USA
S.Smirnov,MathematicsSection,UniversityofGeneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,NY,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Austria
Errico Presutti
Scaling Limits in
Statistical Mechanics
and Microstructures in
Continuum Mechanics
With23Figures
ProfessorErricoPresutti
DipartimentodiMatematica
UniversitàdiRomaTorVergata
ViadellaRicercaScientifica1
00133Roma
Italy
[email protected]
ISBN978-3-540-73304-1 e-ISBN978-3-540-73305-8
DOI10.1007/978-3-540-73305-8
TheoreticalandMathematicalPhysicsISSN1864-5879
LibraryofCongressControlNumber:2008933708
MathematicsSubjectClassification(2000):82B26,49Q20,74A15
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Preface
IbeganthisbookabouteightyearsagoafterAntonioDeSimone,StephanLuckhaus
and Stefan Müller asked me to give a series of lectures on statisticalmechanicsat
theMaxPlanckInstituteinLeipzig.Iwrotesomenotesandaftermanyattemptsto
makethemmorereadableabookfinallycameout.
Thewayacontinuumdescriptionemergesfromatomisticmodelsisanintriguing
andfascinatingsubject,whichisbehindmostofmyscientificlife,inparticularthe
analysis of large scale phenomena in statistical mechanics and their mathematical
formulation in terms of thermodynamic and hydrodynamic limits. The theory has
remarkably progressed in the last decades with contributions from many different
areasofmathematicsandphysics,andwhenaskedtogiveinLeipziganoverview
ofthestateoftheartIacceptedwithgreatpleasure.
Thereisthereversedirectionaswell,wherestartingfromacontinuumdescrip-
tion after successive blow-ups we find microstructures and the prodromes of an
underlying microscopic world. Even though the two directions from microscopics
tomacroscopicsandviceversaareinprinciplesymmetric,mathematicaltechniques
andideashaveproceededquiteseparately.Idiscoveredduringmylecturesthatthere
wasagreatinterestintheaudience,whosebackgroundwasmostlyinanalysisand
continuummechanics,tolearnmethodsandproceduresofstatisticalmechanics.At
the same time it became clear to me that notions and theories developed in con-
tinuum mechanics and PDEs had direct implications on my research in statistical
mechanics and, to state it succinctly, I felt excited by the idea of building bridges
betweenthetwoareas,andthisbookiscertainlypartofsuchefforts.Sincethebook
is meant for both communities, it is written with the presumption that the readers
may not be expert on all topics; thus the analysis starts from the beginning with
parts which are rather elementary and open to younger researchers entering in the
field. Statistical mechanics is exemplified in Part I of the book in the context of
theIsingmodel,toavoidtechnicalproblemsaboutunboundednessofthevariables.
PartIIisdevotedtothemesoscopictheory,whichispresentedbystudyinganon-
local version of the classical scalar Ginzburg–Landau functional, namely the L–P
freeenergyfunctionalintroducedbyLebowitzandPenroseintheiranalysisofKac
potentials.PartIIIismorespecialistic;itshowshowvariationalmethodscharacter-
istic of the mesoscopictheory can be used to implementthe Pirogov–Sinaitheory
ofphasetransitionsinlatticeandcontinuummodelswithKacpotentials.
Thechoiceofworkingonparticularmodels(Ising,theL–Pfunctional)reflects
the didactic purposes of the presentation as well as the taste of the author. I have
usedpartsofthebookforlecturesinschoolsandfortheses.Thevarioustopicscan
be easily singled out to be used for courses or lectures as I have tried to make the
partsnottoostronglycorrelated.Chapter1isanintroductionessentiallybasedon
some survey lectures I gave in the last years, and it hopefully gives a first idea of
flavorandcontentofthebook,withoutenteringintotoomanydetails.
ErricoPresutti, ScalingLimitsinStatisticalMechanicsandMicrostructuresinContinuum v
Mechanics,©Springer2009
vi Preface
PartIisaboutthestatisticalmechanicsoftheIsingmodel.Besidesthebasicel-
ementsofthetheoryIhavealsoincludedamoreadvancedpartonthestructureof
theDLR(Dobrushin,LanfordandRuelle)measuresasacorollaryoftheRohlinthe-
oryofconditionalprobabilitiesforLebesguemeasures,whichisexplainedinsome
detail.Thisisaverybeautifulandinstructivepieceofmathematics;ithasbeenfun-
damentalinmyeducation,andforthisreasonIamfondofitandfeltthatIhadto
insertitinthebook.UsingthetheoryofDLRmeasuresasatechnicaltool,Ihave
thenshownhowtheBoltzmannhypothesisthattheentropyisproportionaltothelog
ofthenumberofstatesallowsonetoderivethethermodynamicpotentials.Withthe
help of DLR theory, it is possible to implement Cramer’s large deviation methods
to prove the existence of the thermodynamic potentials. This is not the traditional
wayfollowedinstatisticalmechanics,butithastheadvantagetounderlineconnec-
tionswithotherfieldslikeprobabilityandinformationtheory.AllthisisinChap.2.
With Chap. 3 begins the analysis of phase transitions. Still, in the context of the
IsingmodelIdiscussherethebasictheoremsaboutexistenceandnon-existenceof
phasetransitions,inparticularthe“Peierlsargument”andthe“Dobrushinunique-
ness theorem,” which have fundamental importance in the whole book. Chapter 4
isaboutmeanfieldandKacpotentials;herescalings,coarsegrainingandfreeen-
ergyfunctionalsappearforthefirsttime:thisisthebeginningofthebridgetowards
mesoscopic and continuum theories. Chapter 5 is about stochastic dynamics; it is
in a sense a detourfrom themain line,but dynamicsenters toomuch intothe mi-
croscopic and macroscopic theories that I could not leave it completely out of the
book. To give a flavor of the research in this field I have presented a derivation
of the macroscopic limit evolution for Glauber dynamics with Kac potentials, but
Ihavealsoinsertedastillelementarypartwherestochasticitypersistsinthelimit,
discussingspinodaldecompositionandtunnelingforthemeanfieldinteraction.
Part II is devoted to the mesoscopic theory. For readers whose main interest is
in continuum mechanics this could be where to start the book with a “smooth in-
troductiontowardsstatisticalmechanics.”Thepresentationisinfactself-contained;
motivationsforthechoiceofthefreeenergyfunctionalscomefromstatisticalme-
chanics,butifthereaderacceptsthefunctionalsasprimitivenotions,thenreferences
toPartIarenotnecessary.Chapter6startswithaderivationofthethermodynamic
potentials,which,inthecontextofthemesoscopictheory,amountstothestudyof
somevariationalproblemswithconstraints.TheanalysisparallelstheoneinChap.2
fortheIsingmodelanditiscertainlyinstructivetoseethetwoinperspective.Ithen
considerdynamicsstudyinganon-localversionoftheAllen–Cahnequationrelated
totheL–Pfreeenergyfunctional.Asinreaction–diffusionequations,propertieslike
the“BarrierLemma”andtheComparisonTheoremareproved.Theyarethenused
to study “large deviations” which, in the mesoscopic theory, refer to estimates of
thefreeenergycostofexcursionsawayfromequilibrium.Herecontoursareintro-
ducedandPeierlsestimatesareproved,inanalogywiththecorrespondingnotions
andresultsintheIsingmodel.Asmentionedbefore,thischaptercouldbeseenas
anintroductiontostatisticalmechanicsandPartIcouldbereadrightafterthis.In
Chap. 7 a first application of large deviation estimates are presented by studying
surfacetensionandWulffproblemsfortheL–Pfunctional.Thebasicnotionshere
Preface vii
areGammaconvergenceandgeometricmeasuretheory,whichinfactplayafunda-
mentalrole;yetsomeoftheirbasictheoremsarehererecalledwithoutproofs.Inthe
originalplanofthebookIhadinmindtoinsertinPartIIIthestatisticalmechanics
analogueinthecontextoftheLMPparticlemodelintroducedbyLebowitz,Mazel
andPresuttitostudyphasetransitionsinthecontinuum,butthebookwasalready
toolongandIgaveup.Chapter8concludestheanalysisofthesurfacetensionfor
theL–Pfreeenergyfunctionalwiththestudyoftheshapeoftheinterface(instan-
ton)intheonedimensionalcase.Theinstantonistheminimizerofthefreeenergy
overprofilesconstrainedtoapproachtheplusandminusequilibriumvaluesatplus
andminusinfinity,itisthereforeablowupoftheinterfacebetweenthetwoequi-
librium phases. Existence and properties of the instanton including the dynamical
∞
onesareprovedinthischapter;inparticular,spectralgapestimatesinaL setting
usinganextensionoftheDobrushinuniquenesstheoryofChap.3,whichinvolves
Vaserstein distance and couplings. I had also planned a chapter about motion by
curvaturebutdroppeditforreasonsofspace(maybenexttime...).
PartIIIismorespecialistic,andithasbeenwrittenwithseveralaims.Onewasto
showhowideasandmethodsofthemesoscopictheorycanfindapplicationsinsta-
tisticalmechanics.Chapter9showsthatthelargedeviationestimatesofChap.6for
theL–PfunctionalcanbeusedtoprovethePeierlsestimatesintheIsingmodelwith
ferromagneticKacpotentialsandhencetheoccurrenceofphasetransitionsind≥2
dimensions.Iliketheresult,becauseitgivesanexampleofthepowerofKacideasin
implementingthevanderWaalstheoryofliquid–vaporphasetransitions.Thecare-
fulanalysisoftheL–PfunctionalinPartIIallowsonetocarryouttheKacprogram
withoutactuallytakingthescalinglimitasγ →0(rangeoftheinteractiontoinfin-
ity)anditshowsthatthemeanfieldphasediagramisagoodapproximationofthe
truephasediagramifγ issmall.Coarsegraining,blockspins,effectivehamiltonian,
andrenormalizationgroupideasappearnaturallyatthisstage.ForbrevityIdonot
discuss the specific structure of DLR measures also because they are examined in
Chaps.11and12inthemorecomplexcontextoftheLMPmodel.Theanalysisof
the Ising model is made simple by the spin flip symmetry, which maps the minus
and plus phases one into the other. The Pirogov–Sinai theory is an important step
in statistical mechanics which provides methods for deriving Peierls estimates on
contours when the symmetry is absent. The original theory was designed to study
perturbationsofthegroundstatesatsmallpositivetemperaturesandmypurposein
the bookis to discuss howthe theory can be adaptedto study perturbationsof the
minimizersofthefreeenergyfunctionalatsmallγ.Insteadofaddingtermstothe
IsinghamiltoniantobreakthespinflipsymmetryIhavethoughtitmoreinformative
toconsidertheLMPparticlemodel.ThisisasystemofpointparticlesinRd which
interact via Kac potentials and which in its mean field approximation has a phase
transitionintoaplusanda minusstatewithdistinctparticledensities.Thereis no
symmetrybetweenthetwophasesandtheanalysisofthesmallγ perturbationsof
theminimizersrequirestheuseofthePirogov–Sinaitheory,whichispresentedin
alldetailinChaps.10and11.InChapter12Ihaveaddedacharacterizationofthe
DLRmeasuresfortheLMPmodelwhichisnotintheexistingliterature.Forreasons
of space I have dropped the derivation of the Gibbs phase rule in LMP, for which
viii Preface
Irefertotheliterature.AttheendofPartsI,IIandIIIthereareshortsectionswith
referencesandnotes.Inthesubjectindexthereaderwillfindalistofthemostused
symbols.
EventhoughIappearastheauthorofthebookIamcertainlynottheonlyone.
Allthisistheoutcomeofmanydiscussionswithfriendsandcolleagues,andparts
are taken from lectures and then modified with the help and the comments of the
audience.Ihavejusttriedtoreorganizeallthat,andtheresultthereforeisnotonly
mine. In particular, the program to study Kac potentials keeping the scaling para-
meterγ fixedwasoriginallyconceivedandthencarriedoutwithMarzioCassandro
towhomIamespeciallyindebted.Dynamicsanditsmacroscopiclimitsareinstead
mostlyrelatedtomyworks withAnnaDeMasi whoalsohelpedmealotinwrit-
ingthisbook.GiovanniBellettiniexplainedtomesomeofthefascinatingideasof
DeGiorgiandhelpedmetoapproachthesubjectfromthesideofthemacroscopic
theory.
The influence of the Moscow school is evident in the book and more generally
in the way mathematical physics looks today, which is, I believe, largely due to
thecontributionsoftheSovietschoolandthisobviouslyreflectsinthebookwhere
thenamesofDobrushinandSinaiarerecurrent.Ilearnedalotfromthemandnot
only mathematics. There are topics which are not yet ready, in particular a whole
big chapter about elastic bodies where there are some intriguing ideas of Stephan
Luckhauswhichwearetryingtodevelop.ButthisisforthefutureandIwouldlike
toconcludethisprefacebymentioningJoelLebowitzwhohasbeenformeateacher
andafriendandifthereissomethinggoodinthisbookthecreditiscertainlyhis.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Themacroscopictheory . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Themesoscopictheory . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Themicroscopictheory . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Kacpotentialsandthemesoscopiclimit. . . . . . . . . . . . . . . 10
1.5 Notesforthereader . . . . . . . . . . . . . . . . . . . . . . . . . 12
PartI StatisticalMechanicsofIsingsystems
2 ThermodynamiclimitintheIsingmodel . . . . . . . . . . . . . . . . 17
2.1 FinitevolumeGibbsmeasures . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Spinconfigurations,phasespace . . . . . . . . . . . . . . 18
2.1.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 FinitevolumeGibbsmeasures. . . . . . . . . . . . . . . . 21
2.1.5 AconsistencypropertyofGibbsmeasures . . . . . . . . . 23
2.1.6 Randomboundaryconditions . . . . . . . . . . . . . . . . 25
2.1.7 StructureoffinitevolumeGibbsmeasures . . . . . . . . . 27
2.2 ThermodynamiclimitandDLRmeasures . . . . . . . . . . . . . . 28
2.2.1 DLRmeasures . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 ThermodynamiclimitsofGibbsmeasures . . . . . . . . . 31
2.2.3 PurestatesandextremalDLRmeasures . . . . . . . . . . 33
2.2.4 SimplicialstructureofDLRmeasures. . . . . . . . . . . . 35
2.2.5 Sigmaalgebraatinfinity . . . . . . . . . . . . . . . . . . . 36
2.2.6 Purephases,phasetransitions . . . . . . . . . . . . . . . . 38
2.2.7 Ergodicdecomposition . . . . . . . . . . . . . . . . . . . 39
2.3 Boltzmannhypothesis,entropyandpressure . . . . . . . . . . . . 42
2.3.1 Anexamplefrominformationtheory . . . . . . . . . . . . 42
2.3.2 Boltzmannhypothesis . . . . . . . . . . . . . . . . . . . . 43
2.3.3 Thermodynamiclimitofthepressure . . . . . . . . . . . . 48
ix
x Contents
2.3.4 Thermodynamiclimitoftheentropy . . . . . . . . . . . . 50
2.3.5 Equivalenceofensembles . . . . . . . . . . . . . . . . . . 54
2.3.6 Propertiesofconvexfunctions. . . . . . . . . . . . . . . . 55
2.3.7 ConcavityoftheBoltzmannentropy . . . . . . . . . . . . 56
2.3.8 Variationalprinciplesandequivalenceofensembles . . . . 61
2.3.9 Avariationalprincipleformeasures. . . . . . . . . . . . . 64
2.4 ThermodynamicsandDLRmeasures . . . . . . . . . . . . . . . . 67
2.4.1 Canonicalensembleandfreeenergy . . . . . . . . . . . . 67
2.4.2 Tangentfunctionalstothepressure . . . . . . . . . . . . . 71
2.4.3 Largedeviations . . . . . . . . . . . . . . . . . . . . . . . 72
3 ThephasediagramofIsingsystems . . . . . . . . . . . . . . . . . . . 75
3.1 Generalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Absenceofphasetransitionsinonedimensions . . . . . . 76
3.1.2 Phasetransitionsatlowtemperatures . . . . . . . . . . . . 78
3.1.3 Absenceofphasetransitionsathightemperatures . . . . . 83
3.2 Dobrushinuniquenesstheory.I.Vasersteindistance . . . . . . . . 88
3.2.1 CouplingsandVasersteindistance. . . . . . . . . . . . . . 88
3.2.2 TheDobrushinmethodforconstructingcouplings . . . . . 90
3.2.3 Totalvariationdistance . . . . . . . . . . . . . . . . . . . 93
3.2.4 Equationsforthefirstmoments . . . . . . . . . . . . . . . 95
3.2.5 DecaypropertiesoftheGreenfunction . . . . . . . . . . . 96
4 Meanfield,KacpotentialsandtheLebowitz–Penroselimit . . . . . . 99
4.1 Themeanfieldmodel . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Thefinitevolume,meanfieldfreeenergy . . . . . . . . . . 100
4.1.2 Theinfinitevolumemeanfieldfreeenergy . . . . . . . . . 101
4.1.3 TheMaxwellequalarealaw . . . . . . . . . . . . . . . . . 103
4.1.4 Themeanfieldpressure . . . . . . . . . . . . . . . . . . . 105
4.1.5 Thermodynamicsofmeanfield . . . . . . . . . . . . . . . 105
4.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2 Kacpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.1 TheLebowitz–Penroselimit . . . . . . . . . . . . . . . . . 107
4.2.2 Blockspins,coarsegraining . . . . . . . . . . . . . . . . . 109
4.2.3 VariationalproblemsandtheL–Pfunctional . . . . . . . . 112
4.2.4 TheL–Pfunctionalasalargedeviationratefunction . . . . 114
4.2.5 ProofofTheorem4.2.2.1 . . . . . . . . . . . . . . . . . . 115
4.2.6 Summaryandfinalcomments . . . . . . . . . . . . . . . . 117
5 StochasticDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1 TheGlaubersemigroup . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1 Spinflipdynamics . . . . . . . . . . . . . . . . . . . . . . 119
5.1.2 Invariantmeasures . . . . . . . . . . . . . . . . . . . . . . 121
5.1.3 Glauberspinfliprates . . . . . . . . . . . . . . . . . . . . 122
5.1.4 Meanfielddynamics . . . . . . . . . . . . . . . . . . . . . 124
5.1.5 Scalinglimitinmeanfielddynamics . . . . . . . . . . . . 126
Contents xi
5.1.6 LimitevolutionofdynamicswithKacpotentials . . . . . . 127
5.2 Markovprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.1 ThespinflipMarkovchains . . . . . . . . . . . . . . . . . 130
5.2.2 ThespinflipMarkovprocess . . . . . . . . . . . . . . . . 131
5.2.3 Poissonprocessesandexponentialtimes . . . . . . . . . . 133
5.2.4 Realizationsofthemeanfieldprocess. . . . . . . . . . . . 137
5.3 Persistenceofstochasticityinthemacroscopiclimit . . . . . . . . 137
5.3.1 Spinodaldecomposition . . . . . . . . . . . . . . . . . . . 137
5.3.2 Largedeviationsinthemeanfielddynamics . . . . . . . . 142
5.3.3 Ratefunctionoflargedeviations . . . . . . . . . . . . . . 143
5.3.4 Costfunctionalandoptimalcostproblems . . . . . . . . . 146
5.4 ComplementstoSect.5.3 . . . . . . . . . . . . . . . . . . . . . . 149
5.4.1 Selfsimilargrowth. . . . . . . . . . . . . . . . . . . . . . 149
5.4.2 Proofof(5.3.1.2) . . . . . . . . . . . . . . . . . . . . . . 152
5.5 NotesandreferencetoPartI. . . . . . . . . . . . . . . . . . . . . 153
PartII MesoscopicTheory,non-localfunctionals
6 Non-local,freeenergyfunctionals . . . . . . . . . . . . . . . . . . . . 157
6.1 Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.1 Thebasicaxioms . . . . . . . . . . . . . . . . . . . . . . 158
6.1.2 Example1.TheGinzburg–Landaufunctional. . . . . . . . 159
6.1.3 Example2.TheL–Pfunctional . . . . . . . . . . . . . . . 160
6.1.4 GeneralizationsoftheL–Pfunctional . . . . . . . . . . . . 163
6.1.5 Example3.TheLMPmodel. . . . . . . . . . . . . . . . . 165
6.1.6 Thermodynamicpotentials . . . . . . . . . . . . . . . . . 165
6.2 PropertiesoftheL–Pfreeenergyfunctional . . . . . . . . . . . . 168
6.2.1 Positivityandlowersemi-continuity . . . . . . . . . . . . 169
6.2.2 Gradientflows,thenon-localL–Pevolution . . . . . . . . 170
6.2.3 Partialdynamics . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.4 Localmeanfieldequations . . . . . . . . . . . . . . . . . 172
6.2.5 Freeenergydissipationforpartialdynamics . . . . . . . . 173
6.2.6 Limitpointsoforbitsforpartialdynamics . . . . . . . . . 174
6.2.7 BarrierLemma . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2.8 Freeenergydissipationininfinitevolumedynamics . . . . 176
6.2.9 Limitpointsoforbitsininfinitevolumedynamics . . . . . 177
6.2.10 ComparisonTheorem . . . . . . . . . . . . . . . . . . . . 178
6.3 Localequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.3.1 Adefinitionoflocalequilibrium . . . . . . . . . . . . . . 182
6.3.2 Timeinvarianceoflocalequilibriumensembles . . . . . . 184
6.3.3 Minimizationoflocalequilibriumprofiles . . . . . . . . . 186
6.4 Largedeviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.4.1 Themulti-canonicalconstraint . . . . . . . . . . . . . . . 188
6.4.2 Extensivebounds . . . . . . . . . . . . . . . . . . . . . . 189
6.4.3 TopologicalpropertiesofD((cid:3))-sets . . . . . . . . . . . . . 191
6.4.4 Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . 192