Scaling Limits of Interacting Particle Systems

452 Pages·1999·11.502 MB·English

by  Claude Kipnis, Claudio Landim (auth.)

Grundlehren der
mathematischen Wissenschaften
320
A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chern B. Eckmann P. de Ia Harpe
H. Hironaka F. Hirzebruch N. Hitchin
L. Hörmander M.-A. Knus A. Kupiainen
J. Lannes G. Lebeau M. Ratner D. Serre
Ya.G. Sinai N. J. A. Sloane J.Tits
M. Waldschmidt S. Watanabe
Managing Editors
M. Berg er J. Coates S. R. S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Claude Kipnis
Claudio Landim
Scaling Limits
of Interacting
Particle Systems
Springer
Claude Kipnist
Claudio Landim
Instituto de Matematica Pura e Aplicada
Estrada Dona Castorina 110
22460-320 Rio de Janeiro, Brasil
and
CNRS UPRES-A 6085
Universite de Rouen
F-76128 Mont Saint Aignan Cedex, France
Library of Congress Cataloging-in-Publication Data
Kipnis, CI aude, 1949-1993
Scaling limits of interacting partide systems 1 Claude Kipnis,
Claudio Landim.
p. cm. - (Grundlehren der mathematischen Wissenschaften ; 320)
Indudes bibliographical references and index.
ISBN 978-3-642-08444-7 ISBN 978-3-662-03752-2 (eBook)
DOI 10.1007/978-3-662-03752-2
1. Scaling laws (Statistic al physics) 2. Probabilities.
3. Markov processes. 4. Hydrodynamics--Mathematics.
5. Mathematical physics. 1. Landim, Claudio. II. Tide.
III. Series.
QC174.85.S34L36 1999
532' .5--dc21
Mathematics Subject Classification (1991): 60-02, 60K35, 82A05
ISSN 0072-7830
ISBN 978-3-642-08444-7
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© Springer-Verlag Berlin Heidelberg 1999
Originally published by Springer-Verlag Berlin Heidelberg New York in 1999
Softcover reprint of the hardcover lSt edition 1999
Cover design: MetaDesign plus GmbH, Berlin
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SPIN: 10631667 41/3143-5 4 3 21 o Printed on acid-free paper
A minha filha Anna
Aos meus pais Raul e Regina
Preface
The idea of writing up a book on the hydrodynamic behavior of interacting particle
systems was born after a series of lectures Claude Kipnis gave at the University
of Paris 7 in the spring of 1988. At this time Claude wrote some notes in French
that covered Chapters 1 and 4, parts of Chapters 2, 5 and Appendix 1 of this
book. His intention was to prepare a text that was as self-contained as possible. lt
would include, for instance, all tools from Markov process theory (c f. Appendix 1,
Chaps. 2 and 4) necessary to enable mathematicians and mathematical physicists
with some knowledge of probability, at the Ievel of Chung (1974), to understand
the techniques of the theory of hydrodynamic Iimits of interacting particle systems.
In the fall of 1991 Claude invited me to complete his notes with him and
transform them into a book that would present to a large audience the latest
developments of the theory in a simple and accessible form. To concentrate on the
main ideas and to avoid unnecessary technical difficulties, we decided to consider
systems evolving in finite lattice spaces and for which the equilibrium states are
product measures. To illustrate the techniques we chose two well-known particle
systems, the generalized exclusion processes and the zero-range processes. We also
conceived the book in such a manner that most chapters can be read independently
of the others.
Here are some comments that might help readers find their way. At the end
of each chapter a Iist of references is given, possible or proved extensions are
discussed, a brief survey of some related topics that could not be covered here is
provided.
To present the main concepts and the principal goals of the theory in the
siruplest possible context, in Chapter 1 we prove the hydrodynamic behavior of
a system with no interaction among particles, more precisely for a Superposition
of independent random walks. We recommend this chapter for someone who has
never studied the subject before. We explain there the need for a microscopic and a
macroscopic time scale, the natural time renormalization in order to observe non
trivial hydrodynamic phenomena for symmetric and asymmetric processes and the
concepts of local equilibrium, conservation of local equilibrium, hydrodynamic
behavior of interacting particle systems and hydrodynamic equations.
The purpose of Chapters 2 and 3 is to review the main properties of gener
alized exclusion processes and zero range processes and to introduce some weak
VIII Preface
formulations of local equilibrium. These two chapters can be skipped by someone
familiar with interacting particle systems.
In Chapter 4 we start to present the main steps and the main tools in a general
proof of the hydrodynamic behavior of interacting particle systems. To illustrate
the approach we consider symmetric simple exclusion processes. The proof of the
hydrodynamic behavior of these systems turns out to be very simple because for
symmetric simple exclusion processes the density field is closed under the action
of the generator. We discuss there the natural topological spaces to consider and
we state some tightness results that are used throughout the book.
In Chapter 5 we explain the entropy method in the context of zero-range pro
cesses. Here the density field is no Ionger closed under the action of the generator
and one needs to project local fields on the density field. This is made possible
by the one and two block estimates that rely on bounds on the entropy and the
Dirichlet form obtained through the examination of the time evolution of the en
tropy. The one and two block estimates are one of the most important results of
this book. Most of the subsequent chapters depend on these estimates.
In Chapter 6 we present the relative entropy method, an alternative approach to
prove the hydrodynamic behavior of systems whose hydrodynamic equation admits
smooth solutions. This is a quite simple method to derive the hydrodynamic Iimit
of an interacting particle system. Its development led to the investigation of the
Navier-Stokes corrections to the hydrodynamic equations. This chapter can be
read independently of the others, we just need the one block estimate.
In Chapter 7 we extend to reversible nongradient systems, i.e., to processes in
which the instantaneous current cannot be written as the difference of a continuous
function and its translation, the entropy method presented in Chapter 5. The proof
of the hydrodynamic behavior of nongradient systems relies on a sharp estimate
for the spectral gap of the generator of the process restricted to finite cubes and
on the characterization of closed and exact forms of f::(ld. The need for a sharp
estimate on the spectral gap should be emphasized because it is completely hidden
in the proof and because one of the main open questions in the field is that of
obtaining a proof of the hydrodynamic behavior of nongradient systems that does
not use any information on the size of the gap. The sharp estimate on the spectral
gap and the characterization of closed and exact forms of Nzd are contained in
Appendix 3. There we also prove some estimates on the largest eigenvalue of
small perturbations of Markov generators.
In Chapter 8 we prove the hydrodynamic behavior of asymmetric attractive
processes in arbitrary dimensions. This chapter can also be read independently
of the others, we just use the one block estimate. The proof relies on the theory
of measure-valued entropy solutions of hyperbolic equations and on the entropy
inequalities presented in Appendix 2.
In Chapter 9 we show how to derive the conservation of local equilibrium from
a law of !arge numbers for local fields in the case of attractive systems. This result
together with the methods presented in Chapter 8 for asymmetric processes and in
Chapters 5, 6 and 7 for symmetric processes allow us to prove the conservation
Preface IX
of local equilibrium for attractive systems. This chapter can be read independently
of the previous ones.
In Chapter 10 we prove a large deviation principle for the density field by
showing that the one and two block estimate are in fact superexponential. In
Chapter 11 we investigate the equilibrium fluctuations of the density field. The
nonequilibrium fluctuations remain a mainly open problem, though there exist
some one-dimensional results.
In Appendix 1 we present all the tools of Markov processes used in the book.
In Appendix 2 we derive, from the local central Iimit theorem for independent
and identically distributed random variables, estimates on the distance between
the finite marginals of canonical and grand canonical equilibrium measures. We
also prove some results on large deviations needed in Chapter 10, fix the termi
nology and recall some weil known results about weak solutions of hyperbolic
and parabolic quasi-linear partial differential equations.
In September 1993, when Claude died, Chapters 1, 2, 4, 5 and parts of Ap
pendix 1 were ready and I decided to conclude alone our original project. I just
made one important modification: the language. All those who met Claude, cer
tainly remernher his incredible facility for learning languages. Even though he
could speak and write in perfect English, the natural language in which to write
a scientific book at the end of the twentieth century, Claude insisted on writing
this book in French, in some sense our common native language. After he died in
1993, I decided, in order to maximise the potential readership, present and future,
to switch to English. I hope Noemi will forgive me.
I take this opportunity to express my thanks to all our collaborators and col
leagues from whom we learned most of the techniques presented here and to all
those who read parts of the book for their comments and their encouragement.
I particularly acknowledge the influence on us of Pablo Ferrari, Antonio Galves,
Gianni Jona-Lasinio, Tom Liggett, Stefano Olla, Errico Presutti, Raghu Varadhan
and Horng-Tzer Yau. I would like also to thank Ellen Saada for her careful reading
of most of the chapters and for her uncountable suggestions and Olivier Benois
for his helpful support with all the subtleties of TJjX.
Campos dos Goytacazes, April 1998. Claudio Landim
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. An Introductory Example: Independent Random Walks . . . . . . . . . 7
1.1 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Loca1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Hydrodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Equivalence of Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2. Some Interacting Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Some Remarks on the Topology of Nzd and M 1(Nzd) . . . . . . . . 21
2.2 Simple Exclusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Zero Range Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Generalized Exclusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Attractive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Zero Range Processes in Infinite Volume . . . . . . . . . . . . . . . . . . . 38
3. Weak Formulations of Local Equilibrium . . . . . . . . . . . . . . . . . . . . . 41
4. Hydrodynamic Equation of Symmetrie Simple Exclusion
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Topology and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 The Hydrodynamic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5. An Example of Reversible Gradient System: Symmetrie Zero
Range Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Proof of the Replacement Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 The One Block Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 The Two Blocks Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 A L2 Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.7 An Energy Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
XII Contents
6. The Relative Entropy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1 Weak Conservation of Local Equilibrium . . . . . . . . . . . . . . . . . . . 116
6.2 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7. Hydrodynamic Limit of Reversible Nongradient Systems . . . . . . . . 141
7.1 Replacing Currents by Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 An Integration by Parts Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Nongradient Large Deviation Estimates . . . . . . . . . . . . . . . . . . . . . 154
7.4 Centtal Limit Theorem Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.5 The Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7. 7 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8. Hydrodynamic Limit of Asymmetrie Attractive Processes . . . . . . . . 191
8.1 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.2 An Entropy Inequality at Microscopic Level . . . . . . . . . . . . . . . . . 208
8.3 Law of Large Numbers for the Empirical Measure . . . . . . . . . . . 218
8.4 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9. Conservation of Local Equilibrium for Attractive Systems . . . . . . . 231
9.1 Replacement Lemma for Attractive Processes . . . . . . . . . . . . . . . . 233
9.2 One Block Estimate Without Time Average . . . . . . . . . . . . . . . . . 235
9.3 Conservation of Local Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 246
9.4 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10. Large Deviations from the Hydrodynamic Limit . . . . . . . . . . . . . . . 257
10.1 The Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10.2 Weakly Asymmetrie Simple Exclusion Processes . . . . . . . . . . . . . 263
10.3 A Superexponential Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.4 Large Deviations Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.5 Large Deviations Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10.6 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11. Equilibrium Fluctuations of Reversible Dynamics . . . . . . . . . . . . . . 287
11.1 The Boltzmann-Gibbs Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11.2 The Martingale Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
11.3 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
11.4 Generalized Omstein-Uhlenbeck Processes . . . . . . . . . . . . . . . . . . 307
11.5 Comments and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309