From Newton to Boltzmann: Hard Spheres and Short-range Potentials

149 Pages·2014·1.224 MB·English

by  Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier

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Isabelle Gallagher
Laure Saint-Raymond
Benjamin Texier
From Newton to
Boltzmann:
Hard Spheres and
Short-range Potentials
Authors:
Isabelle Gallagher Laure Saint-Raymond
Institut Mathématiques de Jussieu (UMR 7586) Université Pierre et Marie Curie and
Université Paris-Diderot (Paris 7) Département de Mathématique et Applications
Bâtiment Sophie Germain, Case 7012 Ecole Normale Superieure
75205 Paris Cedex 13 45 Rue d'Ulm
France 75230 Paris Cedex 05
France
E-mail: [email protected]
E-mail: [email protected]
Benjamin Texier
Institut Mathématiques de Jussieu (UMR 7586)
Université Paris-Diderot (Paris 7)
Bâtiment Sophie Germain, Case 7012
75205 Paris Cedex 13
France
E-mail: [email protected]
2010 Mathematics Subject Classification (Primary; secondary): 35Q20; 35Q70
Key words: Boltzmann equation, particle systems, propagation of chaos, BBGKY hierarchy, hard spheres,
clusters, collision trees
ISBN 978-3-03719-129-3
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9 8 7 6 5 4 3 2 1
Preface
Thesubjectofthismonographistheappearanceofirreversibilityingasdynamics.
Atamolecularlevel,thedynamicsisNewtonian. Inparticular,itisreversible,in
contrastwithobservationsatamacroscopiclevel.
In1872,Boltzmannintroducedtheequation
.B/ @tf Cv(cid:2)rxf DQ.f;f/;
where x 2 Rd represents position and v 2 Rd velocity, for the probability den-
sity f.t;x;v/ known as the distribution function of the gas. The bilinear collision
operator Q is related to a jump process in the velocity variable. The dynamics of
the Boltzmann equationlocally preservesmass, momentumand energy, as does the
Newtonianmicroscopicdynamics.Inaddition,theBoltzmannequationadmitsaLya-
punov functional, known as the entropy, which is nondecreasing along trajectories.
Thisisafeatureofanirreversibledynamics.
The specific question that we address in this monograph is the relationship be-
tweentheNewtondynamicsforasystemofparticlesandtheBoltzmanndynamics.
ApartialanswerisgiveninOscarLanford’s1975theorem[35], whichaccounts
forsomeimportantintuitionsofBoltzmann[9]:
(cid:3) equation (B) should be obtained as a limit when the number of particles be-
comeslarge. InBoltzmann’swords: Thevelocitydistributionofthemolecules
isnotmathematicallyexactaslongasthenumberofmoleculesisnotassumed
tobemathematicallyinfinitelylarge.
(cid:3) equation (B) predicts only the most probable behavior. In particular, it does
not accountfor trajectories along the Newtonian flow which have decreasing
entropy: In nature, the tendency is to pass from the least likely state to the
morelikely. [...] ThesecondprincipleinThermodynamicsappearstherefore
asaprobabilitytheorem.
(cid:3) acentralquestioninthederivationofequation(B)istheindependenceofele-
mentaryparticles: From nowonwe shallspecificallyassumethatthemotion
istotallydisorganized,eitherasanensembleoratamolecularlevel,andthat
itremainssoindefinitely.
Lanford’s theorem states that the distribution function of a system of N particles,
whichareinteractingwithoneanotherbyelasticcollisionsandareinitiallyindepen-
dentand smoothly distributed, convergesto the solution of the Boltzmann equation
(B)inthe limitN ! 1;ifthe characteristiclengthofinteraction" simultaneously
goesto0intheBoltzmann-GradscalinglimitN"d(cid:2)1 DO.1/.
AstrikingpointinLanford’stheoremisthatitpartiallyinvalidatesthethirdintu-
itionofBoltzmann: theindependenceisrigorouslyestablishedinthelimit,underthe
mereassumptionthatitholdsinitially.
vi
Themainlimitationinthetheoremisthattheconvergenceisprovedtoholdonly
on small time intervals, in which typically only a small number of collisions per
particletakeplace.
Asweshallsee,trajectoriesthatarenotaccountedforintheBoltzmanndynamics
involve recollisions, meaning interactions between particles which have previously
interactedinthepast(directlyorindirectly). Suchtrajectoriesviolateindependence.
The strategy of Lanford was then to decompose the dynamics in terms of collision
treesandprovethat
(cid:3) withprobabilityconvergingto1,collisionstreesarefinite,and
(cid:3) withprobabilityconvergingto1,recollisionsdonothappeninfinitetrees.
It seems however that the arguments used in the literature to establish the second
pointwerenotentirelycorrect,sothatatsomepointtheproofshouldbecompleted.
The aim ofthis monographis to providesucha completionoftheproofofLan-
ford’s theorem, in a self-contained manner. In addition, building on the important
contribution of King [31], the convergence result is extended to systems of parti-
cles interacting pairwise via compactly supported potentials satisfying a convexity
assumption. Wealsodiscussindepththenotionofindependence.Inthehard-sphere
case,preciseboundsinallstepsoftheproofenableustoobtainarateofconvergence.
Weinsistonthefactthatthestrategyoftheproofisbynomeansnew. Themain
novelty here is the detailed study of trajectories involving recollisions. This is the
keypointthatallowstoprovetheterm-by-termconvergenceresultinthecorrelation
seriesexpansion.
PartIgivessomecontext: wediscusslow-densitylimits,recallsomeofthemain
landmarks in the vast literature concerning the Boltzmann equation, and state the
maintheoremsprovedinthismonograph.
InPartIIwefocusonthehard-spherecase. WefirstderivetheBBGKYhierarchy
associatedwiththeLiouvilleequation,andprovethatitiswell-posedonashorttime
interval,uniformlyinthenumberofparticles. Thenweturntothenotionofindepen-
dence,whichiscentralinLanford’stheorem. Finallywegiveapreciseconvergence
statementoftheBBGKYhierarchytotheBoltzmannhierarchy. Theconvergenceto
the Boltzmann equation then appears as the particular case of tensor products. We
finallypresentthesalientfeaturesoftheproof.
PartIIIisdevotedtothecaseofparticleinteractionsproducedbyacompactlysup-
ported potential. We first study the scattering operator associated with two-particle
interactions, and then derive the associated BBGKY hierarchy. This derivation is
rendereddelicatebythe factthatsimultaneousinteractionsoflarge numbersofpar-
ticles mayoccur. Only pairwiseinteractionscontributeto thedynamicsin the limit,
however,andboundssimilarto theonesin thehard-spherecasearederived. A pre-
cisestatementofconvergencetowardsthelimitingBoltzmannhierarchyisgiven,and
astrategyofproofispresented.
Part IV presents the proofs of both convergenceresults (hard spheres and short-
rangepotential). Thefactthatpotentialinteractionsarenon-localproducesonlymi-
nordifferencesbetween the proofs. The study oftrajectories involvingrecollisions,
which deviate substantially from the Boltzmann trajectories, is performed in detail.
vii
Inparticular,weprovideexplicit(semi-explicit,inthecaseofapotential)boundson
their size. As a consequence, in the hard-sphere case a rate of convergence can be
obtained. Alistofopenproblemsconcludesthetext.
We thank Jean Bertoin, Thierry Bodineau, Dario Cordero-Erausquin, Laurent
Desvillettes,Franc¸oisGolse,Ste´phaneMischler,Cle´mentMouhotandRobertStrain
for many helpful discussions on topics addressed in this text. We are particularly
grateful to Mario Pulvirenti, Chiara Saffirio and Sergio Simonella for explaining to
ushowcondition(8.3.1)makespossibleaparametrizationofthecollisionintegralby
thedeflectionangle(seeChapt.8).
Finally we thank the anonymous referee for helpful suggestions to improve the
manuscript.
Paris,May2013 IsabelleGallagher
LaureSaint-Raymond
BenjaminTexier

Contents
I Introduction 1
1 Thelowdensitylimit 2
1.1 TheLiouvilleequation . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Meanfieldversuscollisionaldynamics . . . . . . . . . . . . . . . . . 4
1.3 TheBoltzmann-Gradlimit . . . . . . . . . . . . . . . . . . . . . . . 6
2 TheBoltzmannequation 7
2.1 Transportandcollisions . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Boltzmann’sH-theoremandirreversibility . . . . . . . . . . . . . . . 8
2.3 TheCauchyproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Short-timeexistenceofcontinuoussolutions. . . . . . . . . . . 10
2.3.2 Fluctuationsaroundsomeglobalequilibrium. . . . . . . . . . 11
2.3.3 Renormalizedsolutions. . . . . . . . . . . . . . . . . . . . . . 12
3 Mainresults 14
3.1 LanfordandKing’stheorems . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Backgroundandreferences . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Newcontributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II The caseofhardspheres 19
4 MicroscopicdynamicsandBBGKYhierarchy 20
4.1 TheN-particleflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 TheLiouvilleequationandtheBBGKYhierarchy . . . . . . . . . . . 22
4.3 WeakformulationofLiouville’sequation . . . . . . . . . . . . . . . 23
4.4 TheBoltzmannhierarchyandtheBoltzmannequation . . . . . . . . . 27
5 UniformaprioriestimatesfortheBBGKYandBoltzmannhierarchies 30
5.1 Functionalspacesandstatementoftheresults . . . . . . . . . . . . . 30
5.2 Mainstepsoftheproofs . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Continuityestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.4 Someremarksonthestrategyofproof . . . . . . . . . . . . . . . . . 38
6 Statementoftheconvergenceresult 39
6.1 Quasi-independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1.1 AdmissibleBoltzmanndata. . . . . . . . . . . . . . . . . . . . 39
6.1.2 Conditioning. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.1.3 CharacterizationofadmissibleBoltzmanninitialdata. . . . . . 41
x Contents
6.2 Mainresult:
ConvergenceoftheBBGKYhierarchytotheBoltzmannhierarchy . . 46
6.2.1 Statementoftheresult. . . . . . . . . . . . . . . . . . . . . . 46
6.2.2 AbouttheproofofTheorem8: outlineofChapter7andPartIV. 47
7 Strategyoftheproofofconvergence 49
7.1 Reductiontoafinitenumberofcollisiontimes . . . . . . . . . . . . . 50
7.2 Energytruncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.3 Timeseparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.4 Reformulationintermsofpseudo-trajectories . . . . . . . . . . . . . 54
III The caseofshort-range potentials 57
8 Two-particleinteractions 58
8.1 Reducedmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Scatteringmap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.3 Scatteringcross-sectionandtheBoltzmanncollisionoperator . . . . . 64
8.3.1 Scatteringcross-section. . . . . . . . . . . . . . . . . . . . . . 64
8.3.2 Boltzmanncollisionoperator. . . . . . . . . . . . . . . . . . . 67
9 TruncatedmarginalsandtheBBGKYhierarchy 68
9.1 Truncatedmarginals . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.2 WeakformulationofLiouville’sequation . . . . . . . . . . . . . . . 70
9.3 Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4 Collisionoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.5 Mildsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.6 ThelimitingBoltzmannhierarchy . . . . . . . . . . . . . . . . . . . 79
10 Clusterestimatesanduniformaprioriestimates 81
10.1 Clusterestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.2 Functionalspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 Continuityestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10.4 UniformboundsfortheBBGKYandBoltzmannhierarchies . . . . . 90
11 Convergenceresultandstrategyofproof 91
11.1 Admissibleinitialdata . . . . . . . . . . . . . . . . . . . . . . . . . 91
11.2 ConvergencetotheBoltzmannhierarchy . . . . . . . . . . . . . . . . 93
11.3 ReductionsoftheBBGKYhierarchy,andpseudo-trajectories . . . . . 94
IV Termwiseconvergence 98
12 Eliminationofrecollisions 99
12.1 Stabilityofgoodconfigurationsbyadjunctionofcollisionalparticles . 99
12.2 Geometriclemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 102