FROM BEING TO BECOMING TIME AND COMPLEXITY IN THE PHYSICAL SCIENCES i- ILYA PRIGOGINE ^•'^ 0^^ J. ta /- ^^ } FROM BEING TO BECOMING J^-' 1^^^^^ ^ Pre-Columbian calcite model of a temple, dated before 300 B.C., from the state of Guerrero in IVIexico. Its height is 10 cm. Private collection. FROM BEING TO BECOMING TIME AND COMPLEXITY IN THE PHYSICAL SCIENCES ILYA PRIGOGINE Free University of Brussels and The University of Texas at Austin CB W. H. FREEMAN AND COMPANY New York Photographs of the Belousov-Zhabotinskii reaction at the beginning of each chapter are by Fritz Goro. Sponsoring Editor: Peter Renz Designer: Robert Ishi Production Coordinator: William Murdock Illustration Coordinator: Cheryl Nufer Artists: John and Jean Foster Compositor: Santype International Limited Printer and Binder: The Maple-Vail Book Manufacturing Group Library of Congress Cataloging in Publication Data Prigogine, Ilya. From being to becoming: time and complexity in the physical sciences. Bibliography: p. Includes index. 1. Space and time. 2. Irreversible processes. 3. Physics—Philosophy. I. Title. QC173.59.S65P76 500.2'01 79-26774 ISBN 0-7167-1107-9 ISBN 0-7167-1108-7 pbk. Copyright © 1980 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America 9 10 II 12 13 14 15 VB 5 4 3 2 10 8 9 To my colleagues and friends in Brussels and Austin whose collaboration has made this work possible CONTENTS PREFACE xi Chapter 1 Introduction: Time in Physics 1 The Dynamical Description and Its Limits 1 The Second Law of Thermodynamics 5 Molecular Description of Irreversible Processes Time and Dynamics 12 Parti THE PHYSICS OF BEING 15 Chapter 2 Classical Dynamics 19 Introduction 19 Hamiltonian Equations of Motion and Ensemble Theory 21 Operators 27 Equilibrium Ensembles 29 Integrable Systems 29 Ergodic Systems 33 Dynamical Systems neither Integrable nor Ergodic 38 Weak Stability 43 Chapter 3 Quantum Mechanics 47 Introduction 47 Operators and Complementarity 49 Quantization Rules 51 Time Change in Quantum Mechanics 56 Ensemble Theory in Quantum Mechanics 59 Schrodinger and Heisenberg Representations 62 Equilibrium Ensembles 64 The Measurement Problem 65 Decay of Unstable Particles 67 Is Quantum Mechanics Complete? 70 Part II THE PHYSICS OF BECOMING 73 Chapter 4 Thermodynamics 77 Entropy and Boltzmann's Order Principle 77 Linear Nonequilibrium Thermodynamics 84 Thermodynamic Stability Theory 90 Application to Chemical Reactions 94 CONTENTS Chapter 5 Self-Organization 103 Stability, Bifurcation, and Catastrophes 103 Bifurcations: The Brusselator 109 A Solvable Model for Bifurcation 116 Coherent Structures in Chemistry and Biology 120 Ecology 123 Concluding Remarks 126 Chapter 6 Nonequilibrium Fluctuations 131 The Breakdown of the Law of Large Numbers 131 Chemical Games 135 Nonequilibrium Phase Transitions 139 Critical Fluctuations in Nonequilibrium Systems 142 Oscillations and Time Symmetry Breaking 143 Limits to Complexity 145 Effect of Environmental Noise 147 Concluding Remarks 154 Part III THE BRIDGE FROM BEING TO BECOMING 151 Chapter 7 Kinetic Theory 155 Introduction 155 Boltzmann's Kinetic Theory 159 Correlations and the Entropy of Rejuvenation 165 Gibbs Entropy 170 The Poincare-Misra Theorem 171 A New Complementarity 173 Chapter 8 The Microscopic Theory of Irreversible Processes 179 Irreversibility and the Extension of the Fomalism of Classical and Quantum Mechanics 179 A New Transformation Theory 181 Construction of the Entropy Operator and the Transformation Theory: The Baker Transformation 187 Entropy Operator and the Poincare Catastrophe 191 Microscopic Interpretation of the Second Law of Thermodynamics: Collective Modes 194 Particles and Dissipation: A Non-Hamiltonian Microworld 197 Chapter 9 The Laws of Change 201 Einstein's Dilemma 201 Time and Change 204 Time and Entropy as Operators 206 Levels of Description 210 Past and Future 212 An Open World 214 CONTENTS APPENDIXES 217 A. Time and Entropy Operators for the Baker Transformation 219 B. Resonances and Kinetic Description 232 C. Entropy, Measurement, and the Superposition Principle in Quantum Mechanics 241 Pure States and Mixtures 241 Entropy Operator and Generator of Motion 242 The Entropy Superoperator 245 D. Coherence and Randomness in Quantum Theory 249 Operators and Superoperators 249 Classical Commutation Rules 251 Quantum Commutation Rules 252 Concluding Remarks 254 REFERENCES 257 NAME INDEX 263 SUBJECT INDEX 267 Come, press me tenderly upon your breast But not too hard, for fear the glass might break This is the way things are: the World Scarcely suffices for the natural. But the artificial needs to be confined. GOETHE, Faust, Part II PREFACE This book is about time. I would like to have named it Time, the Forgotten Dimension, although such a title might surprise some readers. Is not time incorporated from the start in dynamics, in the study of motion? Is not time the very point of concern of the special theory of relativity ? This is certainly true. However, in the dynamical description, be it classical or quantum, time enters only in a quite restricted way, in the sense that these equations are invariant with respect to time inversion, f -» — f. Although a specific type of interaction, the so-called superweak interaction, seems to violate this time symmetry, the violation plays no role in the problems that are the subject of this book. As early as 1754, d'Alembert noted that time appears in dynamics as a mere "geometrical parameter" (d'Alembert 1754). And Lagrange, more than a hundred years before the work of Einstein and Minkowski, went so far as to call dynamics a four-dimensional geometry (Lagrange 1796). In this view, future and past play the same role. The world lines, the trajectories, followed by the atoms or particles that make up our universe can be traced toward the future or toward the past. This static view of the world is rooted in the origin of Western science (Sambursky 1963). The Milesian school, of which Thales was one of the most illustrious proponents, introduced the idea of a primordial matter closely related to the concept of conservation of matter. For Thales, a single substance (such as water) forms the primordial matter; all changes PREFACE in physical phenomena, such as growth and decay, must therefore be mere illusions. Physicists and chemists know that a description in which past and future play the same role does not apply to all phenomena. Everybody observes that two liquids put into the same vessel generally diffuse toward some homogeneous mixture. In this experiment, the direction of time is essential. We observe progressive homogenization, and the one-sidedness of time becomes evident in the fact that we do not observe spontaneous phase separation of the two mixed liquids. But for a long time such phenomena were excluded from the fundamental description of physics. All time-oriented processes were considered to be the effect of special, "improbable" initial conditions. At the beginning of this century, this static view was almost unani­ mously accepted by the scientific community, as will be seen in Chapter 1. But we have since been moving away from it. A dynamical view in which time plays an essential role prevails in nearly all fields of science. The concept of evolution seems to be central to our understanding of the physical universe. It emerged with full force in the nineteenth century. It is remarkable that it appeared almost simultaneously in physics, biology, and sociology, although with quite different specific meanings. In physics it was introduced through the second law of thermodynamics, the celebrated law of increase of entropy, which is one of the main subjects of this book. In the classical view, the second law expressed the increase of molecular disorder; as expressed by Boltzmann, thermodynamic equilibrium corresponds to the state of maximum "probability." However, in biology and sociology, the basic meaning of evolution was just the opposite, describing instead transformations to higher levels of complexity. How can we relate these various meanings of time—time as motion, as in dynamics; time related to irreversibility, as in thermodynamics; time as history, as in biology and sociology? It is evident that this is not an easy matter. Yet, we are living in a single universe. To reach a coherent view of the world of which we are a part, we must find some way to pass from one description to another. A basic aim of this book is to convey to the reader my conviction that we are in a period of scientific revolution—one in which the very position and meaning of the scientific approach are undergoing re- PREFACE xiii appraisal—a period not unlike the birth of the scientific approach in ancient Greece or of its renaissance in the time of Galileo. Many interesting and fundamental discoveries have broadened our scientific horizon. To cite only a few: quarks in elementary particle physics; strange objects like pulsars in the sky; the amazing progress of molecular biology. These are landmarks of our times, which are especially rich in important discoveries. However, when I speak of a scientific revolution, I have in mind something different, something perhaps more subtle. Since the beginning of Western science, we have believed in the "simplicity" of the microscopic—molecules, atoms, elementary particles. Irreversibility and evolution appear, then, as illusions related to the complexity of collective behavior of intrinsically simple objects. This conception—historically one of the driving forces of Western science— can hardly be maintained today. The elementary particles that we know are complex objects that can be produced and can decay. If there is simplicity somewhere in physics and chemistry, it is not in the microscopic models.