Chaos, Order.

Michael Bushev Institute of Solid State Physics

Bulgarian Academy of Sciences

|*5 World Scientific Singapore • New Jersey • L Singapore • New Jersey • London • Hong Kong

CONTENTS

Preface vii

I SYNERGY OF SCIENCES 1 THE SUBJECT OF SYNERGETICS 3

1.1. The Name 3 1.2. A Brief History 4 1.3. Self-Organization Phenomena 5 1.4. Dissipative Structures 5 1.5. Autowave Processes 6 1.6. Self-Organization, Cybernetics, Autopoiesis 7 1.7. Catastrophe Theory 8 1.8. On the New Concepts 11 1.9. Theory of Bifurcations 12 References 14

2 SYSTEM, STRUCTURE, SELF-ORGANIZATION 17

2.1. Once More on the Subject of Synergetics 17 2.2. Systems and Systematic Approach 18 2.3. Structure and Organization 19 2.4. Self-Organization 21 2.5. Symmetry of Causes and Symmetry of Consequences 25 References and Comments 29

3 SELF-ORGANIZATION P H E N O M E N A 31

3.1. Equilibrium and Nonequilibrium Phase Transitions 31

xi

CONTENTS

3.2. Equilibrium Structures and Phase Transitions 31 3.2.1. Phases and phase transitions 31 3.2.2. First-order and second-order phase transitions 32 3.2.3. Critical fluctuations 32 3.2.4. Symmetry, organization, self-organization 34 3.2.5. Order parameter and control parameter 35 3.2.6. Long-range and short-range order 36 3.2.7. Bose-Einstein condensation 38

3.3. Nonequilibrium Phase Transitions 38 3.3.1. Hydrodynamic instabilities 39 3.3.2. Laser light 44 3.3.3. Nonstationary Josephson effect 47 3.3.4. Gunn effect 48 3.3.5. Auiovibrations and autowaves 49 3.3.6. Soliions 50 3.3.7. Chemistry: Belousov-Zhabotinski reactions 55 3.3.8. Biology: embryogenesis and morphogenesis 58 3.3.9. Biorhythms (biological clocks) 60 3.3.10. Bioimpulses 61 3.3.11. Ecology: population waves 61 3.3.12. Other fields 62 3.3.13. Summary of the self-organization phenomena 63

References and Comments 63

THE ROLE OF THE MEDIUM 67 4.1. Susceptibility 67 4.2. Deterministic Media 68 4.3. Stability, Critical Fluctuations, Critical Slowing Down 69 4.4. Bifurcations 70 4.5. Catastrophes 72 4.6. Random Media (External Noise) 73 References and Comments 75

LEVELS OF DESCRIPTION A N D APPROACHES TO SELF-ORGANIZATION 77 5.1. Levels of Description 78 5.2. The Irreversibility Paradox 80 5.3. Reduced Description 84 Acknowledgments 88 References and Comments 88

CONTENTS xiii

II CHAOS, ORDER, EVOLUTION 6 CHAOS, ORDER, SYMMETRY 93

6.1. Chaos and Cosmos 93 6.2. Order and Symmetry 94 6.3. Chance and Chaos, Non-Deterministic Chaos 95 6.4. What is Primary — Chaos or Order? 97 6.5. The Fermi-Pasta-Ulam Paradox 99 6.6. Deterministic Chaos 101 6.7. Simple and Strange Attractors 103 6.8. Fractal Dimension 106 6.9. The Butterfly Effect and the Long-Term Prognoses ПО 6.10. The Perfection 110 References and Comments 112

7 EVOLUTION OF THE EVOLUTIONARY VIEWS 115 7.1. The Cosmogonic Myths 115 7.2. Plato's Dialogue Timaeus 116 7.3. The Static World of Parmenidus 117 7.4. Creationism and Vitalism 117 7.5. Immanuel Kant's Waves 118 7.6. The Quasi-Static World of Laplace-Einstein 119 7.7. Thermodynamics 121 7.8. Thermodynamic Evolution 123 7.9. Entropy and the Systematic View 125 7.10. The Arrow of Time 126 7.11. The Heat-Death Paradox 127 7.12. Biological Evolution 129 7.13. Evolutionary Processes 131 7.14. Cosmic Calendar 131 7.15. Physical Evolution 133 7.16. Review of the Evolutionary Views 135 References and Comments 136

8 EVOLUTION E Q U A T I O N S 139 8.1. Linear and Nonlinear Waves 139 8.2. Dispersion and Dissipation of Waves 140 8.3. "Pure" Nonlinearity 142 8.4. Nonlinearity and Dissipation 144

xiv CONTENTS

8.5. Nonlinearity and Dispersion 145 8.6. Other Evolution Equations Describing Solitons 147 8.7. General Dynamic Evolution Equation 148 8.8. Stochastic Evolution Equations 149 References 150

III THEORETICAL APPROACHES TO THE PROCESSES OF SELF-ORGANIZATION

Physics and the Mystery of Life 153

9 T H E R M O D Y N A M I C APPROACH 155

9.1. The Le Chatelier-Braun Principle 156 9.2. Stationary States 157 9.3. The Principle of Onsager 158 9.4. Entropy Production 160 9.5. Theorem of Prigogine 162 9.6. The Le Chatelier-Braun Principle for Stationary States . . . 164 9.7. Types of Thermodynamic Structures 165 9.8. Nonlinear Thermodynamics 166 9.9. Theorem of Glansdorff-Prigogine 167 9.10. Summary of the Thermodynamic Approach 169 References and Comments 169

10 THE DYNAMIC APPROACH , 171

10.1. Phase Portrait of a Dynamic System 171 10.2. Autocatalytic Point Models of Lotka and Volterra 173 10.3. The Model "Brusselator" 175 10.4. The Oregonator Model 177 10.5. Turing Model 179 10.6. Pulse Running Along a Nerve Filament 182 10.7. Heat Structures 183 10.8. Physical Systems 186 10.9. Fluctuations 187 10.10. The Role of Chance 187 References and Notes 188

11 THE SYNERGETIC APPROACH 191

11.1. A Short "Guide" to the Ideas of the Approach 191

CONTENTS XV

11.2. Random Processes 192 11.3. Stochastic Evolution 195 11.4. Stochasticity in Dynamics (Channel I) 196 11.5. The Fokker-Planck Equation 197 11.6. The Role of the Fluctuations (Channel II) 198 11.7. The Slaving Principle (Channel III) 200 11.8. The Evolution Equation for the Order Parameter

(Cascade IV) 202 11.9. Nonequilibrium Phase Transitions 204 11.10. Summary of the Synergetic Approach 205 11.11. Chaos — Invalidity of the Slaving Principle 207 11.12. Concluding Remarks 208 References 208

IV THE NEW ALLIANCE

12 METAEVOLUTION 213

12.1. The Epistemological Problem of II. Poincare 213 12.2. Phenomena, Laws, Invariance 214 12.3. "End" of Physics 215 12.4. Metaevolution 216 12.5. "Law without Law" 218 References 219

13 METAMORPHOSIS OF SCIENCE 221

13.1. Complexity and Simplicity 221 13.2. The Myths of Science 223 13.3. The Rulership Myth 224 13.4. The Synergetic Exclusion Principle 225 13.5. A World of Nonlinearities 226 References 227

14 MAN'S DIALOGUE WITH NATURE 229

14.1. Frankenstein versus Pygmalion 229 14.2. The Two Cultures 230 14.3. The Problem of Time 231 14.4. Actors and Spectators 231 References and Comments 233

XVI CONTENTS

APPENDICES A COMPLEMENTARITY A N D IRREVERSIBILITY 237

A.l. Complementarity in Quantum Mechanics 237 A.2. Complementarity beyond Physics 238 A.3. Historicity (Evolution) in Sciences 239 A.4. Complementarity of Dynamic and Thermodynamic

Description 240 References and Comments 240

В INFORMATION A N D SELF-ORGANIZATION 243 B.l. What do Living Organisms Feed on? 243 B.2. Entropy and Information 245 B.3. On Biological Order 248 B.4. Quality (Value) of Information 249 B.5. Organisms as Aperiodic crystals 251 B.6. Informational Aspects of Self-Organization 252 B.7. Algorithmic Complexity 253 B.8. Symmetry and Information 255 References and Comments 257

С P H A S E SPACE 259

References and Comments 261

N A M E INDEX 263

SUBJECT INDEX 269