The Nature of Irreversibility: A Study of Its Dynamics and Physical Origins

349 Pages·1985·17.67 MB·English

by  Henry B. Hollinger

THE NATURE OF IRREVERSIBILITY
THE UNIVERSITY OF WESTERN ONTARIO
SERIES IN PHILOSOPHY OF SCIENCE
A SERIES OF BOOKS
IN PHILOSOPHY OF SCIENCE, METHODOLOGY,
EPISTEMOLOGY, LOGIC, HISTORY OF SCIENCE,
AND RELATED FIELDS
Managing Editor
ROBERT E. BUTTS
Dept. ofP hilosophy, University of Western Ontario, Canada
Editorial Board
JEFFREY BUB, University of Western Ontario
L. JONATHAN COHEN, Queen's College, Oxford
WILLIAM DEMOPOULOS, University of Western Ontario
WILLIAM HARPER, University of Western Ontario
JAAKKO HINTIKKA, Florida State University, Tallahassee
CLIFFORD A. HOOKER, University of Newcastle
HENRY E. KYBURG, JR., University ofRoch~ter
AUSONIO MARRAS, University of Western Ontario
JURGEN MITTELSTRASS, Universitiit Konstanz
JOHN M. NICHOLAS, University of Western Ontario
GLENN A. PEARCE, University of West em Ontario
BAS C. VAN FRAASSEN,Princeton University
VOLUME 28
THE NATURE OF
IRREVERSIBILITY
A Study of Its Dynamics and Physical Origins
by
HENRY B. HOLLINGER
Department of Chemistry, Rensselaer Polytechnic Institute
and
MICHAEL JOHN ZENZEN
Department of Philosophy, Rensselaer Polytechnic Institute
D. REIDEL PUBLISHING COMPANY
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER/TOKYO
libory of Congreu Cataloging in Public~tion Data
lIollingl:l, Henry B., 1933-·
The n~turc of irreversibility.
(The Univ~rsity of Westcrn Ontario scril:s in philo!lOphy of scicn"e : v. 28)
Bibliography: p.
Includcs indexes.
I. Irreversible PIOC<:SSCS. 2. Fluid dynamics. 3, Statistical mechanics.
I. Zenten. MichacJJohn, 1945- 11. Title. HI. Series.
OC174.l7.176H65 1985 530.1 85-18280
ISBN-13: 978-94-010-8897-8 e-ISBN-13: 978-94-009-5430-4
DOl: 10.1007f978-94-009-S430-4
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PREFACE
A dominant feature of our ordinary experience of the
world is a sense of irreversible change: things lose form,
people grow old, energy dissipates. On the other hand, a
major conceptual scheme we use to describe the natural
world, molecular dynamics, has reversibility at its core.
The need to harmonize conceptual schemes and experience
leads to several questions, one of which is the focus of
this book. How does irreversibility at the macroscopic
level emerge from the reversibility that prevails at the
molecular level?
Attempts to explain the emergence have emphasized
probability, and assigned different probabilities to the
forward and reversed directions of processes so that one
direction is far more probable than the other. The conclu
sion is promising, but the reasons for it have been
obscure. In many cases the aim has been to find an explana
tion in the nature of probability itself. Reactions to that
have been divided: some think the aim is justified while
others think it is absurd.
Other accounts of irreversibility have appealed to
averaging procedures used in probabilistic calculations, to
stochastic flaws in the underlying mechanical model, to the
large numbers of molecules involved, to jammed correlations,
unseen perturbations, hidden variables, uncertainty
principles, and cosmological factors. While acknowledging
these attempts as important articulations of basic ideas in
statistical mechanics and other theories, we feel that they
do not meet the irreversibility paradox head-on. They do
not explain the origin of irreversibility as it occurs in
the common, everyday, natural behavior of observable
materials. We think the natural macroscopic irreversibility
can be understood only by examining the predicament of the
systems involved. Irreversibility is not an intrinsic
feature of any system, however large or small, quantized or
a
not, fast or slow, Whatever. Rather it is consequence of
a predicament which can be identified in terms of the ratio
between the typical times for internal changes and the
typical times between external changes. Any mechanical
system can perform reversibly in some predicament. Any
predicament can allow some systems to be reversible and
force others to be irreversible.
v
vi PREFACE
The issue for molecular dynamics is to understand the
patterns of behavior for isolated systems. In that predica
ment, the behavior is reversible. Then the effects of
external changes can be examined to determine where and how
they induce irreversibility. Probability and statistical
mechanics can be useful as tools in these examinations
provided they are applied only to reproducible and pre
dictable phenomena, and are never substituted for the roles
played by molecular dynamics.
What is needed and largely missing in the current
theories about molecular dynamics is a science of "equili
brium plateaus" in the temporal variations of collective
properties. It seems to us that once the phenomenon of
equilibrium is understood in terms of molecular dynamics,
the macroscopic appearance of irreversibility can be under
stood in terms of the frequency of forced withdrawals from
young equilibria. We believe that the paradox of irrevers
ibility can be resolved in a simple, logically clear, and
aesthetically pleasing manner.
In the absence of a science of equilibrium in many
particle systems, we must proceed on empirical knowledge.
For observed fluids the distribution of plateau lengths
seems to be such that we can generally assume a plateau
length is longer than necessary to induce irreversible
behavior. That is our assumption throughout this book.
The assumption is supported by sketchy estimates in our
literature and by what we have found in computer simula
tions of rarefied gases, which should have the shortest
plateaus.
This book is intended primarily for philosophers and
scientists who have a special interest in the interpretive
and foundational problems associated with statistical
mechanics and the dynamics of irreversibility. For those
readers who want to see details about how irreversibility
enters fluid dynamics and particle dynamics and how it is
treated in statistical mechanics, we have included addi
tional chapters which reflect material used to complement
the course here in nonequilibrium statistical mechanics.
In those chapters we have tried to provide a rationale for
the mathematics and to provide enough detail to be nearly
self-contained while at the same time aiming the discussion
at questions about irreversibility.
PREFACE
The description here belies the effort that went into
the study. Both of us had been independently interested in
questions about temporal anisotropy for many years before
we began to collaborate. And we argued for several more
years before we decided to try to gather together What
history and conclusions we could. To our surprise, things
came together and many of our arguments became slight dif
ferences over details. We now want to communicate our
conclusions to other people Who ponder irreversibility, and
beyond that, to engage new participants.
There may be those like Hollinger Who start out
believing that time is merely a parameter for Newtonian
motion. Relativity is merely an alteration in the scheme
and does not argue against the fact that Newton can still
sit in his frame (maybe not even inertial) and expect to
calculate everything that can happen in any frame, includ
ing slowed clocks, rotations, and whatever comes out of any
new theories. And there may be others like Zenzen Who have
the physics background but resist the dogma and want to
leave open certain questions about time in its most general
and abstract aspect. The two of us have been bent toward
each other's view, and we have found agreement on many
details. Hollinger must now admit that time is a bigger
mystery than he assumed in those years, and Zenzen agrees
that Newtonian time, for all its simplicity, does allow for
a wider range of explanation than is generally recognized.
Neither feels that he has reached the end in these studies,
but each feels that he has made some progress.
We are grateful to Professor Robert Butts and other
colleagues Who have offered encouragement and/or critical
comments, all of Which have affected the presentation. We
are grateful to Geri Frank for her masterful editing and
typing. We are grateful to the Philosophy of Science
Association for permission to use parts of our article in
Philosophy of Science, 49 (1982) pp.309-354.
TABLE OF CONTENTS
PREFACE v
1. INTRODUCTION 1
2. THE PARADOXE S 5
2.1 Early Studies of Heat and Attempts to
Formulate Equations of Heat Flow... ......• 5
2.2 Thompson's 1852 Statement on Irreversibility. 7
2.3 Dissipative Processes and Irreversible
Processes Not Yet Distinguished •.. '. ... ... 8
2.4 Statistical Notions Enter Kinetic Theory 9
2.5 Boltzmann Tries to Reduce the Second
Law to Mechanics .. .. .. ... .. .. .... .. . .. .... 11
2.6 The "H" Theorem and Loschmidt's
Reversibility Paradox • . . . . . . . . . .. . . . . . . . . . 12
2.7 The Reversibility Paradox Rediscovered 16
2.8 Boltzmann's Philosophy of Science •. ......• 18
2.9 The Boltzmann-Planck Debate ............... 19
2.10 Ehrenfests and the Problem of
Irreversibility •. . . . . . . . • . . . . . . . .. • . . . .. •. 28
3. THE APPLICATIONS 30
3.1 Transport Rates Determined by Mean
Free Paths • . . . . . . . . • . . .. . .. . . . . . . . •. . . .. . . 31
3.2 Transport Rates Determined by the
Boltzmann Equation ......• '" ". .... . ... ... 35
4. RETURN TO THE PARADOXE S 41
4.1 The Loss of Information 41
4.2 Microscopic Reversibility •................ 41
4.3 The Role of Recent Equilibrium ...•......•. 42
4.4 Molecular Chaos and the BBGKY Theory 43
4.5 Later Developments • ' ....•.•....•.......... 52
5. VARIOUS KINDS OF IRREVERSIBILITY 57
5.1 Inertial Irreversibility 57
5.2 Temporal Irreversibility 60
5.3 Exclusion Irreversibility 64
x TABLE OF CONTENTS
5.4 Mixing the Criteria: Thermodynamic
Irreversibility •. . . . . . . • . . . . . . . . .. . . . . . . 65
5.5 Mixing the Criteria: Paradoxical
Irreversibility •. .... ... ...... ....... ... 67
5.6 Refinements: de Facto and Nomological
Irreversibility • . . . . . . . . . . . . . . . . . . . . . . .• 68
5.7 Statistical Irreversibility: Necessarily
de Facto •........................... '" . 72
6. PROPOSED ORIGINS OF IRREVERSIBILITY 87
6.1 Probabilistic Origins 88
6.2 Mechanical Origins 95
7. THE ORIGIN OF EXCLUSION IRREVERSIBILITY 98
7.1 The Simplest Newtonian Models .... ....... 98
7.2 The Role of Time Scales ., ............ '" 103
7.3 Exclusion and Dissipation ... ...... ...... 106
7.4 The Principle of Recent Equilibrium 110
7.5 A Reflection ... .... ...... .. . ...... ...... 111
8. IRREVERSIBILITY IN FLUID DYNAMICS 113
8. 1 The Fluid Concept .. . .. . .. . .. .. .. . . . .. . . . 113
8.2 Fluid Processes . . . . . . . . . .. . .. . .. . . . . . .. . 116
8.3 Fluid Equations ........ . .... ............ 117
8.4 Fundamental Equations of Change •. ... .... 118
8.5 Stochastic Equations of Change •... ...... 136
8.6 Simple Equations of Flux .. ... ........ ... 140
8.7 Complex Equations of Flux •..... .... ..... 146
8.8 Equations of Equilibrium •............ '" 148
9. IRREVERSIBILITY IN STATISTICAL MECHANICS 159
9.1 The Method of Statistical Mechanics 160
9.2 Generalization to Systems of
Interacting Particles • . . . . . • . . . . . . . .• . . . 178
9.3 Generalization to a Continuum of States 183
9.4 The Liouville Theorem .. '" .... ... ... .•.. 188
9.5 Joining Statistics and Mechanics: The
One-Particle Approximation •• ..•.•....••. 195
9.6 Complex Equations of Flux in the One-
Particle Approximation • . . . . • . . . . . .• . . . . . 214
TABLE OF CONTENTS xi
9.7 The Two-Particle Approximation ........ . 227
9.8 Higher Approximations ................. . 247
10. IRREVERSIBILITY IN QUANTUM STATISTICAL
MECHANICS ........•.......................... 257
10.1 The Schrodinger Equation ............. . 258
10.2 The One-Particle Approximation ....... . 267
10.3 The Two-Particle Approximation ....... . 276
10.4 The Chemical Approximation ........... . 283
11. ON ALTERNATIVE APPROACHES 289
APPENDIX - SOME REFLECTIONS ON TIME AND
TEMPORAL ITY ..................... . 304
NOTES 311
REFERENCES 319
NAME INDEX 331
SUBJECT INDEX 334