Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

307 Pages·2016·3.143 MB·English

by  Adrian Muntean, Jens Rademacher, Antonios Zagaris (eds.)

Lecture Notes in Applied Mathematics and Mechanics
Adrian Muntean
Jens D.M. Rademacher
Antonios Zagaris Editors
Macroscopic and Large
Scale Phenomena:
Coarse Graining,
Mean Field Limits and
Ergodicity
Lecture Notes in Applied Mathematics
and Mechanics
Volume 3
Series editors
Alexander Mielke, Humboldt-Universität zu Berlin, Berlin, Germany
e-mail: [email protected]
Bob Svendsen, RWTH Aachen University, Aachen, Germany
e-mail: [email protected]
Associate editors
Helmut Abels, University of Regensburg, Regensburg, Germany
Marek Behr, RWTH Aachen University, Aachen, Germany
Peter Eberhard, University of Stuttgart, Stuttgart, Germany
Klaus Hackl, Ruhr University Bochum, Bochum, Germany
Axel Klawonn, Universität zu Köln, Köln, Germany
Karsten Urban, University of Ulm, Ulm, Germany
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Adrian Muntean Jens D.M. Rademacher
(cid:129)
Antonios Zagaris
Editors
Macroscopic and Large Scale
Phenomena: Coarse
Graining, Mean Field Limits
and Ergodicity
123
Editors
Adrian Muntean AntoniosZagaris
Department ofMathematics andComputer Faculteit Elektrotechniek, Wiskundeen
Science, Centerfor Analysis, Scientific Informatica
computingandApplications (CASA) Universiteit Twente
Eindhoven University of Technology Enschede,Overijssel
Eindhoven TheNetherlands
TheNetherlands
Jens D.M.Rademacher
Fachbereich 3 - MathematikundInformatik
UniversitätBremen
Bremen
Germany
ISSN 2197-6724 ISSN 2197-6732 (electronic)
Lecture Notesin AppliedMathematics andMechanics
ISBN978-3-319-26882-8 ISBN978-3-319-26883-5 (eBook)
DOI 10.1007/978-3-319-26883-5
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Preface
This book is the offspring of the summer school “Macroscopic and Large Scale
Phenomena: Coarse Graining,Mean Field Limits andErgodicity.”The school was
held in 2012 at the University of Twente, the Netherlands, under the aegis of the
national cluster Nonlinear Dynamics in Natural Systems (NDNS+). We assumed
the initiative to organize that school to bring certain aspects of particle systems
theory to the attention of junior applied mathematicians at large and to applied
analystsinparticular.Tothateffect,themainfocuswasonthemodeling,multiscale
analysis, and continuum limits of particle systems.
This emphasis is reflected in the book at hand, which collects in print an
extensionofthematerialpresentedinthatsummerschool.Eachofthefourchapters
here is based on a set of lectures delivered at the school, yet all authors have
expandedandrefinedtheircontributions.Thefirstthreeofthesediscussmeanfield
limits and (evolutionary) C-convergence as methods to derive macroscopic model
equationsfrommicroscopicdescriptions.Thechapterconcludingthevolumeoffers
a more abstract perspective, discussing the statistical properties of large-scale
dynamics from the standpoint of ergodic and dynamical systems theory. Each
chapterconcernsoneparticularaspectoftheoverallthemeandreflectstheauthor’s
specific viewpoint and focus. We hope that this book can continue to stimulate
interaction between these somewhat separate fields, in the spirit of the school that
generated it.
François Golse delivers a chapter on the dynamics of large particle systems in
the mean field limit. In doing so, he deals with a central issue in statistical
mechanics: the derivation of reduced dynamical descriptions of large systems
composedofidenticalmicroscopicconstituents.Insuchsystems,theeffectofeach
individual particle is weak, but the collective action of the particle ensemble gen-
eratesanontrivialmeanfieldactingoneachandeveryparticle.Anearlyexampleof
such a reduced description is the kinetic theory of gases, which is the mean field
limit of molecular dynamics. According to that theory, the state of a system of a
large number of identical gas molecules is reflected in the statistical behavior of
one typical particle in the system. Golse surveys the most significant tools and
v
vi Preface
methods employed to date to establish mean field limits with mathematical rigor.
Heillustratestheirapplicationactivelybymeansofavarietyofexamplesrootedin
physical reality, such as regularized variants of the Vlasov–Poisson or Vlasov–
Maxwell systems from plasma physics.
Lucia Scardia focuses on the derivation of continuum limits of discrete models
using variational methods. The specific focus is on C-convergence, a more recent
method that has proved very powerful in tackling problems in material science.
Fracture mechanics, spin systems, magneto-mechanics, and dislocations are coun-
ted among the many successes of this approach. A common feature of all these
problems is that good models are known at the discrete level of atoms and dislo-
cations (microscale), but they are missing at the continuum level (macroscale)
which is relevant for engineering applications. The notion of C-convergence
furnishes a mathematically rigorous way to bridge those scales effectively. Scardia
starts with an introduction to this concept, illustrated by concrete examples of
discrete-to-continuum upscalings for simple systems. The latter part of the chapter
deals with a more advanced application in dislocation theory.
Alexander Mielke’s contribution focuses on the multiscale modeling and rig-
orous analysis of generalized gradient systems. Specifically, he examines the new
concept of evolutionary C-convergence for systems driven by a dissipation
mechanismforafunctional,suchasanenergyorentropy.Thefocusinthisworkis
on one-parameter families of such vector fields, where the (small) parameter
quantifies the multiscale structure of the system in question. In this setting, the
central challenge is to describe the limiting behavior of solutions to the detailed
modelintermsofsolutionstoasimplified,appropriatelychosen,macroscopicone.
Inthiscontext,thestaticconceptofC-convergenceprovidesaframeworkinwhich
to understand and perform such upscalings. The presentation connects to physical
and engineering reality by numerous evocative examples relating, for instance, to
periodic homogenization or to the passage from viscous to dry friction.
MartinGöllandEvgenyVerbitskiyconcludethisvolumewithareviewofrecent
advances in a modern branch of ergodic theory. Statistically motivated approaches
lieattheheartofergodictheoryandthestudyofcomplexdynamicalsystems.Their
numerousoffshootshaveembracedadiversearrayofmathematicalfieldsincluding
numbertheory,algebraandgeometry.Therelevantportionofthesummerschoolin
Twente discussed fundamental notions of ergodic theory and demonstrated appli-
cationsofergodicityinnonlineardynamics.Thechapterinthisvolumegoesmuch
further and reviews recent developments in the study of homoclinic points for
certain classes of discrete dynamical systems. From our point of view, the most
striking feature is that these relate certain particle systems to algebraic dynamical
systemsviaergodicpropertiesof,e.g.,latticesconfigurations.Theauthorsmotivate
the abstract notions and theory by numerous examples, in particular, spatial
dynamical systems. Having defined homoclinic points arising through principal
algebraicactions,theyreviewexistingresultsanddemonstratetheuseofthetheory
in computing the entropy and in studying probabilistic properties of these discrete
systems.
Preface vii
WeexpressourgratitudetotheNetherlandsOrganisationforScientificResearch
(NWO), the Dutch Applied Mathematics Institute (3TU.AMI), and the Dutch
Research School in Mathematics (WONDER) for supporting the summer school
financially. We also thank the authors, who kindly agreed to contribute to this
volume, as well as the school participants who proved eager to learn about these
different approaches. Last but not least, we thank the series editors and publishers
for realizing this book.
Eindhoven Adrian Muntean
Bremen Jens D.M. Rademacher
Twente Antonios Zagaris
June 2015
Contents
1 On the Dynamics of Large Particle Systems in the Mean
Field Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
François Golse
1.1 Examples of Mean Field Models in Classical Mechanics . . . . . 3
1.1.1 The Liouville Equation . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 The Vlasov-Poisson System . . . . . . . . . . . . . . . . . . . 5
1.1.3 The Euler Equation for Two-Dimensional
Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.4 The Vlasov-Maxwell System . . . . . . . . . . . . . . . . . . 10
1.2 A General Formalism for Mean Field Limits in Classical
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 The Mean Field Characteristic Flow . . . . . . . . . . . . . . . . . . . 16
1.4 Dobrushin’s Stability Estimate and the Mean Field Limit. . . . . 24
1.4.1 The Monge-Kantorovich Distance. . . . . . . . . . . . . . . 24
1.4.2 Dobrushin’s Estimate. . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 The Mean Field Limit . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.4 On the Choice of the Initial Data . . . . . . . . . . . . . . . 31
1.5 The BBGKY Hierarchy and the Mean Field Limit . . . . . . . . . 35
1.5.1 N-Particle Distributions . . . . . . . . . . . . . . . . . . . . . . 36
1.5.2 Marginal Distributions of Symmetric N-Particle
Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.3 The N-Particle Liouville Equation. . . . . . . . . . . . . . . 41
1.5.4 The BBGKY Hierarchy. . . . . . . . . . . . . . . . . . . . . . 44
1.5.5 The Mean Field Hierarchy and Factorized
Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.6 Chaotic Sequences, Empirical Measures and BBGKY
Hierarchies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6.1 Chaotic Sequences and Empirical Measures . . . . . . . . 56
1.6.2 From Dobrushin’s Theorem to the BBGKY
Hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ix
x Contents
1.7 Further Results on Mean Field Limits in Classical
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.7.1 Propagation of Chaos and Quantitative Estimates . . . . 65
1.7.2 Infinite Hierarchies and Statistical Solutions. . . . . . . . 67
1.7.3 Symmetric Functions of Infinitely Many Variables . . . 74
1.7.4 The Case of Singular Interaction Kernels. . . . . . . . . . 77
1.7.5 From Particle Systems to the Vlasov-Maxwell
System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.8 The Mean Field Problem in Quantum Mechanics . . . . . . . . . . 82
1.8.1 The N-Body Problem in Quantum Mechanics. . . . . . . 82
1.8.2 The Target Mean Field Equation. . . . . . . . . . . . . . . . 87
1.8.3 The Formalism of Density Matrices. . . . . . . . . . . . . . 88
1.9 Elements of Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . 94
1.10 The BBGKY Hierarchy in Quantum Mechanics . . . . . . . . . . . 100
1.10.1 The Quantum BBGKY Hierarchy . . . . . . . . . . . . . . . 101
1.10.2 The Infinite Quantum Mean Field Hierarchy. . . . . . . . 105
1.11 The Mean Field Limit in Quantum Mechanics
and Hartree’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.11.1 Mathematical Statement of the Mean Field Limit . . . . 110
1.11.2 A Tool for Studying Infinite Hierarchies . . . . . . . . . . 113
1.11.3 Application to the Hartree Limit in the Bounded
Potential Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1.12 Other Mean Field Limits in Quantum Mechanics . . . . . . . . . . 122
1.12.1 Derivation of the Schrödinger-Poisson Equation . . . . . 122
1.12.2 Derivation of the Nonlinear Schrödinger Equation. . . . 124
1.12.3 The Time-Dependent Hartree-Fock Equations. . . . . . . 125
1.12.4 Pickl’s Approach to Quantum Mean Field Limits . . . . 130
1.13 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2 Continuum Limits of Discrete Models via C-Convergence. . . . . . . 145
Lucia Scardia
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.2.1 C-Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.2.2 Coerciveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.2.3 Lower Semicontinuity and Relaxation . . . . . . . . . . . . 152
2.3 Discrete Systems: Short-Range Interactions . . . . . . . . . . . . . . 155
2.3.1 One-Dimensional Discrete Setting. . . . . . . . . . . . . . . 155
2.3.2 Nearest-Neighbour Interactions. . . . . . . . . . . . . . . . . 156
2.3.3 Non-convex and Superlinear Interaction. . . . . . . . . . . 162
2.3.4 Lennard-Jones Interactions. . . . . . . . . . . . . . . . . . . . 164