Table Of ContentTexts and
Monographs
in Physics
w. Beiglbock
R. P. Geroch
E. H. Lieb
T. Regge
W. Thirring
Series Editors
Herbert Spohn
Large Scale Dynamics
of Interacting Particles
With 19 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Herbert Spohn
Theoretische Physik, Ludwig-Maximilians-Universitat Munchen,
TheresienstraBe 37, W-8000 Munchen 2, Fed. Rep. of Germany
Editors Elliott H. Lieb
Jadwin Hall
Princeton University
P. O. Box 708
Princeton, NJ 08544-0708, USA
Wolf BeiglbOck Tullio Regge
Institut fur Angewandte Mathematik Istituto di Fisica Teorica
Universitat Heidelberg Universita di Torino
1m Neuenheimer Feld 294 C. so M. d'Azeglio, 46
W-6900 Heidelberg 1, FRG 1-10125 Torino, Italy
Robert P. Geroch Walter Thirring
Enrico Fermi Institute Institut fUr Theoretische Physik
University of Chicago der Universitat Wien
5640 Ellis Ave. Boltzmanngasse 5
Chicago, IL 60637, USA A-1090 Wien, Austria
ISBN-13 :978-3-642-84373-0 e-ISBN-13:978-3-642-84371-6
DOl: 10.1007/978-3-642-84371-6
Library of Congress Cataloging-in-Publication Data. Spohn, Herbert, 1946-Large scale dynamics of interacting
particles 1 Herbert Spohn. p. cm. - (Texts and monographs in physics) Includes bibliographical references and
index.lSBN·13:978-3·642·84373·0(U.S.:alk.paper)l.Many-body problems. 2. Hydrodynamics. I. Title. II. Series.
QC 174.17.P7S66 1991 530.1'44-<\c20 90-27687
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For Sian and Shanti
Preface
The core of the material on large scale dynamics of interacting particles grew
out of courses I taught at the Katholieke Universiteit Leuven, Rutgers Universi
ty, and the Ludwig-Maximilians-Universitat Munchen and out of lectures I
gave at the workshop "Hydrodynamical Behavior of Microscopic Systems" at
the Universita dell'Aquila.
I had the good luck of being helped through difficult ground by many
friends. Amongst them I am deeply indebted to Joel L. Lebowitz. He got me
started. Relatively little would have been achieved without his never-ending
curiosity and insistence on clarity. Furthermore, I gratefully acknowledge the
cooperation of Michael Aizenman, Henk van Beijeren, Carlo Boldrighini,
Jean Bricmont, Paola Calderoni, Brian Davies, Anna DeMasi, Roland
Dobrushin, Detlef Durr, Gregory Eyink, Mark Fannes, Pablo Ferrari, Alberto
Frigerio, Joseph Fritz, Antonio Galves, Shelly Goldstein, Vittorio Gorini,
Reinhard Illner, Claude Kipnis, Joachim Krug, Oscar Lanford, Reinhard Lang,
Joel Lebowitz, Christian Maes, Stefano Olla, George Papanicolaou, Errico
Presutti, Mario Pulvirenti, Fraydoun Rezakhanlou, Hermann Rost, Yasha
Sinai, Yuri Suhov, Domo Szasz, Ragu Varadhan, Andre Verbeure, David Wick,
and Horng-Tzer Yau. The list is somewhat lengthy, perhaps, but besides thanks
I want to make clear that what I will describe is the outcome of a common
scientific enterprise.
I thank Henk van Beijeren and Detlef Durr for careful reading of and com
ments on a previous version. Paola Calderoni and Detlef Durr supplied me
with the proof in Part I, Chapter 8.4 which is most appreciated.
Munchen, May 1991 Herbert Spohn
Contents
Introduction ................................................. .
Scales ..................................................... 1
Outline. .... . . ... . .. . .. . ..... . . ... . .. . ...... ... . . . ... . . . . .. 3
Part I Classical Particles
1. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Newtonian Dynamics ................................... 7
1.2 Boundary Conditions ................................... 10
1.3 Dynamics of Infinitely Many Particles .................... 12
2. States of Equilibrium and Local Equilibrium. . . . . . . . . . . . . . . . . . . 14
2.1 Equilibrium Measures, Correlation Functions .............. 15
2.2 The Infinite Volume Limit .............................. 18
2.3 Local Equilibrium States ................................ 21
2.4 Local Stationarity ...................................... 25
2.5 The Static Continuum Limit ............................ 25
3. The Hydrodynamic Limit...... . ... . . ... . .... . ... . . . .... . . . . . 33
3.1 Propagation of Local Equilibrium ........................ 33
3.2 Hydrodynamic Equations ............................... 34
3.3 The Hard Rod Fluid ................................... 39
3.4 Steady States .......................................... 44
4. Low Density Limit: The Boltzmann Equation .................. 48
4.1 Low Density (Boltzmann-Grad) Limit. . . . . . . . . . .. . . . . . . . . 50
4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories 51
4.3 Convergence of the Scaled Correlation Functions .......... 58
4.4 The Boltzmann Hierarchy ............................... 64
4.5 Time Reversal ......................................... 66
4.6 Law of Large Numbers, Local Poisson .................... 67
4.7 The H-Function ....................................... 70
4.8 Extensions ............................................ 73
X Contents
5. The Vlasov Equation 77
6. The Landau Equation ........................... . . . . . . . . . . . . 83
7. Time Correlations and Fluctuations. . . . . . . . . . .. . . . . . . . . . . . . . . . 85
7.1 Fluctuation Fields ...................................... 85
7.2 The Green - Kubo Formula .............................. 93
7.3 Transport for the Hard Rod Fluid ........................ 96
7.4 The Fluctuating Boltzmann Equation ..................... 99
7.5 The Fluctuating Vlasov Equation ........................ 101
8. Dynamics of a Tracer Particle ................................ 104
8.1 Brownian Particle in a Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.2 The Stationary Velocity Process .......................... 105
8.3 Brownian Motion (Hydrodynamic) Limit. . . . . . . . . . . . . . . . . . 111
8.4 Large Mass Limit ...................................... 119
8.5 Weak Coupling Limit.. .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.6 Low Density Limit ..................................... 133
8.7 Mean Field Limit ...................................... 139
8.8 External Forces and the Einstein Relation ................. 140
8.9 Self-Diffusion ......................................... 142
8.10 Corrections to Markovian Limits. . . . . .. . . . . . . . .. .. . . . . . . . 145
9. The Role of Probability, Irreversibility ........................ 151
Part II Stochastic Lattice Gases
1. Lattice Gases with Hard Core Exclusion ...................... 157
1.1 Dynamics ............................................. 157
1.2 Stochastic Reversibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
1.3 Invariant Measures, Ergodicity, Domains of Attraction.. . . .. 166
1.4 Driven Lattice Gases ................................... 168
1.5 Standard Models ....................................... 171
2. Equilibrium Fluctuations .................................... 175
2.1 Density Correlations and Bulk Diffusion .................. 175
2.2 The Green-Kubo Formula .............................. 177
2.3 Currents .............................................. 181
2.4 The Gradient Condition ................................ 182
2.5 Linear Response, Conductivity ........................... 185
2.6 Steady State Transport.. . .. . . .. . . . . . .. . . . . . . . . . . . . .. . . . . 188
2.7 State of Minimal Entropy Production .................... 190
2.8 Bounds on the Conductivity ............................. 192
Contents XI
2.9 The Field of Density Fluctuations ........................ 193
2.10 Scaling Limit for the Density Fluctuation Field (Proof) ..... 197
2.11 Critical Dynamics ...................................... 209
3. Nonequilibrium Dynamics for Reversible Lattice Gases .......... 212
3.1 The Nonlinear Diffusion Equation ....................... 212
3.2 Hydrodynamic Limit (Proof) ............................ 218
3.3 Low Temperatures ...................................... 229
3.4 Weakly Driven Lattice Gases ............................ 233
3.5 Nonequilibrium Fluctuations ............................ 234
3.6 Local Equilibrium States and Minimal Entropy Production .. 237
3.7 Large Deviations ....................................... 241
4. Nonequilibrium Dynamics of Driven Lattice Gases ............. 252
4.1 Hyperbolic Equation of Conservation Type ................ 252
4.2 Asymmetric Exclusion Dynamics ......................... 257
4.3 Fluctuation Theory ..................................... 259
5. Beyond the Hydrodynamic Time Scale ........................ 262
5.1 Navier-Stokes Correction for Driven Lattice Gases ......... 262
5.2 Local Structure of a Shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
5.2.1 Macroscopic Equation with Fluctuations ............. 267
5.2.2 Shock in a Random Frame of Reference ............. 270
5.2.3 Shock in Higher Dimensions ....................... 273
6. lhlcer Dynamics ........................................... 277
6.1 Two Component Systems ................................ 277
6.2 Tracer Diffusion ....................................... 280
6.3 Convergence to Brownian Motion ........................ 285
6.4 Nearest Neighbor Jumps in One Dimension:
The Case of Vanishing Self-Diffusion ..................... 291
7. Stochastic Models with a Single Conservation Law
Other than Lattice Gases .................................... 296
7.1 Lattice Gases Without Hard Core/Zero Range Dynamics .... 296
7.2 Interacting Brownian Particles ........................... 298
7.3 Ginzburg-Landau Dynamics ............................ 300
8. Non-Hydrodynamic Limit Dynamics .......................... 306
8.1 Kinetic Limit .......................................... 308
8.2 Mean Field Limit ...................................... 312
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 317
List of Mathematical Symbols .................................. 339
Subject Index ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 341
Introduction
Scales
It is amazing how different the world appears on different scales. Let us take as
an example fluid flow, say the river Isar moving at slow pace through the city of
Miinchen. Its flow patterns are governed by the Navier-Stokes equations. These
are deterministic field equations (partial differential equations). If we could
increase the degree of spatial resolution by a factor 10 7, then we would see H20
(and other) molecules moving fairly rapidly in a random like fashion. To
strengthen our imagination of an unfamiliar world, computers are of great help.
It is a striking experience to watch a movie of a molecular dynamics simulation
of a large number of interacting particles governed by Newton's equation of
motion. Particles collide incessantly and move erratically without any particular
aim. How do these particles then manage to organize themselves in such a way
as to form a flow pattern on a large scale? Very crudely the reason is that the
local conservation laws of mass, momentum and energy impose constraints not
immediately visible on the microscopic scale. In what will follow our goal is to
elucidate how from a random motion on the small scale a deterministic motion
on the large scale emerges.
On a somewhat more theoretiCal level there is a further remarkable feature:
To determine the fluid flow we do not have to go all the way back to the
microscopic equations of motion. The large scale dynamics is governed by a set
of autonomous equations. The microscopic interactions appear only indirectly
through the thermodynamics (equation of state) and through the transport
coefficients of the fluid under consideration. We claim, and will explain, that the
autonomous dynamics emerges in a mathematically rigorous fashion through
a scaling limit, i.e. in an idealized limit of infinite scale separation.
In a first course on theoretical physics, or even before, we are taught that the
microscopic motions of particles are given by the solution of a set of ordinary
differential equations. Once the initial condition is prescribed, future and past
are determined. Thus the random motion on the small scale can only be
apparently random. For a few degrees of freedom we have learned in recent
years that no contradiction results. Points in phase space which are initially
close separate exponentially fast under the dynamics (to preserve phase space
volume there must be both exponentially fast contracting and expanding direc
tions). This sensitive dependence on initial conditions produces a dynamical
behavior which rightfully is called random or chaotic. Unfortunately, despite
2 Introduction
intense efforts, for many degrees of freedom deterministic chaos remains a poor
ly understood subject. I am convinced that deterministic chaos on the small
scale is needed in order to have the kind oflarge scale dynamics we actually see.
The good dynamical mixing provides the mechanism through which a system of
many particles approaches and maintains a state close to local equilibrium.
Since we do not know how to prove such a good dynamical mixing for systems
of many particles, our investigations appear to be blocked at a very early stage
indeed. To be able to proceed anyhow we propose two compromises.
i) We simply assume a random motion on the small scale. Particles are then
no longer governed by Newton's equation of motion, but follow some stochastic
dynamics. In this case, we still have the freedom to choose the conservation
laws. For simplicity only the number of particles will be taken to be locally
conserved. The models so obtained are not on the same fundamental footing as
Newtonian particles. However they do describe fairly accurately a large variety
of physical systems of interest. Perhaps the most beautiful example is a suspen
sion consisting of many (1012 fcm 3) polysterene balls immersed in water. The
polystyrene balls are small with a diameter of about 1000 A, but large and heavy
compared to the water molecules. Thus the dynamics of a single ball is given by
Brownian motion. Velocities relax so fast (10 -12 sec) to their thermal distribu
tion that they can safely be ignored as dynamical degrees of freedom. Now,
through chemical processes at their surface the polysterene balls are spontan
eously charged and therefore interact through a screened Coulomb potential.
With a high level of accuracy such suspensions are modelled then theoretically
as a system of interacting Brownian particles, cf. Chap. 7.2 in Part II. We
emphasize that, although random motion on a small scale is assumed, the
problem of the emergence of an autonomous large scale dynamics remains and
has to be analyzed.
ii) We study a system of many particles interacting through a short range
repulsive potential and governed by Newton's equation of motion. The dy
namics is deterministic. Randomness enters only through the initial conditions.
The simplification consists now in considering only the low density regime. The
large scale dynamics is then given by the Boltzmann equation which governs the
density in the one-particle (position and velocity) space. On a heuristic level it is
easy to understand why at low density deterministic chaos cannot be of such
a crucial importance. The time scale appropriate for the Boltzmann equation is
the mean free time. On this time scale every particle undergoes a collision only
once in a while. Every collision causes nearby dynamical trajectories to diverge.
The effect does not add up, however, as long as only a fixed bounded time
interval is under consideration. Of course, for times long compared to the
mean free time, deterministic chaos will become of relevance again. The validity
of the "Stosszahlansatz" can be traced back to the randomness in the initial
conditions as propagated by the collisionless flow. On the other hand the
collisions are responsible for the increase of entropy and for the irreversible
character of the Boltzmann equation.
