Synergetics: An Introduction

Over the past years the field of synergetics has been mushrooming. An ever- increasing number of scientific papers are published on the subject, and numerous conferences all over the world are devoted to it. Depending on the particular aspects of synergetics being treated, these conferences can have such varied titles as "Nonequilibrium Nonlinear Statistical Physics," "Self-Organization," "Chaos and Order," and others. Many professors and students have expressed the view that the present book provides a good introduction to this new field. This is also reflected by the fact that it has been translated into Russian, Japanese, Chinese, German, and other languages, and that the second edition has also sold out. I am taking the third edition as an opportunity to cover some important recent developments and to make the book still more readable. First, I have largely revised the section on self-organization in continuously extended media and entirely rewritten the section on the Benard instability. Sec- ond, because the methods of synergetics are penetrating such fields as eco- nomics, I have included an economic model on the transition from full employ- ment to underemployment in which I use the concept of nonequilibrium phase transitions developed elsewhere in the book. Third, because a great many papers are currently devoted to the fascinating problem of chaotic motion, I have added a section on discrete maps. These maps are widely used in such problems, and can reveal period-doubling bifurcations, intermittency, and chaos.

1. Goal.- 1.1 Order and Disorder: Some Typical Phenomena.- 1.2 Some Typical Problems and Difficulties.- 1.3 How We Shall Proceed.- 2. Probability.- 2.1 Object of Our Investigations: The Sample Space.- 2.2 Random Variables.- 2.3 Probability.- 2.4 Distribution.- 2.5 Random Variables with Densities.- 2.6 Joint Probability.- 2.7 Mathematical Expectation E(X), and Moments.- 2.8 Conditional Probabilities.- 2.9 Independent and Dependent Random Variables.- 2.10 Generating Functions and Characteristic Functions.- 2.11 A Special Probability Distribution: Binomial Distribution.- 2.12 The Poisson Distribution.- 2.13 The Normal Distribution (Gaussian Distribution).- 2.14 Stirling’s Formula.- 2.15 Central Limit Theorem.- 3. Information.- 3.1 Some Basic Ideas.- 3.2 Information Gain: An Illustrative Derivation.- 3.3 Information Entropy and Constraints.- 3.4 An Example from Physics: Thermodynamics.- 3.5 An Approach to Irreversible Thermodynamics.- 3.6 Entropy—Curse of Statistical Mechanics?.- 4. Chance.- 4.1 A Model of Brownian Movement.- 4.2 The Random Walk Model and Its Master Equation.- 4.3 Joint Probability and Paths. Markov Processes. The Chapman-Kolmogorov Equation. Path Integrals.- Sections with an asterisk in the heading may be omitted during a first reading..- 4.4 How to Use Joint Probabilities. Moments. Characteristic Function. Gaussian Processes.- 4.5 The Master Equation.- 4.6 Exact Stationary Solution of the Master Equation for Systems in Detailed Balance.- 4.7 The Master Equation with Detailed Balance. Symmetrization, Eigenvalues and Eigenstates.- 4.8 Kirchhoff’s Method of Solution of the Master Equation.- 4.9 Theorems about Solutions of the Master Equation.- 4.10 The Meaning of Random Processes, Stationary State, Fluctuations, Recurrence Time.- 4.11 Master Equation and Limitations of Irreversible Thermodynamics.- 5. Necessity.- 5.1 Dynamic Processes.- 5.2 Critical Points and Trajectories in a Phase Plane. Once Again Limit Cycles.- 5.3 Stability.- 5.4 Examples and Exercises on Bifurcation and Stability.- 5.5 Classification of Static Instabilities, or an Elementary Approach to Thorn’s Theory of Catastrophes.- 6. Chance and Necessity.- 6.1 Langevin Equations: An Example.- 6.2 Reservoirs and Random Forces.- 6.3 The Fokker-Planck Equation.- 6.4 Some Properties and Stationary Solutions of the Fokker-Planck-Equation.- 6.6 Time-Dependent Solutions of the Fokker-Planck Equation.- 6.6 Solution of the Fokker-Planck Equation by Path Integrals.- 6.7 Phase Transition Analogy.- 6.8 Phase Transition Analogy in Continuous Media: Space-Dependent Order Parameter.- 7. Self-Organization.- 7.1 Organization.- 7.2 Self-Organization.- 7.3 The Role of Fluctuations: Reliability or Adaptibility? Switching.- 7.4 Adiabatic Elimination of Fast Relaxing Variables from the Fokker-Planck Equation.- 7.5 Adiabatic Elimination of Fast Relaxing Variables from the Master Equation.- 7.6 Self-Organization in Continuously Extended Media. An Outline of the Mathematical Approach.- 7.7 Generalized Ginzburg-Landau Equations for Nonequilibrium Phase Transitions.- 7.8 Higher-Order Contributions to Generalized Ginzburg-Landau Equations.- 7.9 Scaling Theory of Continuously Extended Nonequilibrium Systems.- 7.10 Soft-Mode Instability.- 7.11 Hard-Mode Instability.- 8. Physical Systems.- 8.1 Cooperative Effects in the Laser: Self-Organization and Phase Transition.- 8.2 The Laser Equations in the Mode Picture.- 8.3 The Order Parameter Concept.- 8.4 The Single-Mode Laser.- 8.5 The Multimode Laser.- 8.6 Laser with Continuously Many Modes. Analogy with Superconductivity.- 8.7 First-Order Phase Transitions of the Single-Mode Laser.- 8.8 Hierarchy of Laser Instabilities and Ultrashort Laser Pulses.- 8.9 Instabilities in Fluid Dynamics: The Bénard and Taylor Problems.- 8.10 The Basic Equations.- 8.11 The Introduction of New Variables.- 8.12 Damped and Neutral Solutions (R ? Rc).- 8.13 Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations.- 8.14 The Fokker-Planck Equation and Its Stationary Solution.- 8.15 A Model for the Statistical Dynamics of the Gunn Instability Near Threshold.- 8.16 Elastic Stability: Outline of Some Basic Ideas.- 9. Chemical and Biochemical Systems.- 9.1 Chemical and Biochemical Reactions.- 9.2 Deterministic Processes, Without Diffusion, One Variable.- 9.3 Reaction and Diffusion Equations.- 9.4 Reaction-Diffusion Model with Two or Three Variables: The Brusselator and the Oregonator.- 9.5 Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable.- 9.6 Stochastic Model for a Chemical Reaction with Diffusion. One Variable.- 9.7 Stochastic Treatment of the Brusselator Close to Its Soft-Mode Instability.- 9.8 Chemical Networks.- 10. Applications to Biology.- 10.1 Ecology, Population-Dynamics.- 10.2 Stochastic Models for a Predator-Prey System.- 10.3 A Simple Mathematical Model for Evolutionary Processes.- 10.4 A Model for Morphogenesis.- 10.5 Order Parameters and Morphogenesis.- 10.6 Some Comments on Models of Morphogenesis.- 11. Sociology and Economics.- 11.1 A Stochastic Model for the Formation of Public Opinion.- 11.2 Phase Transitions in Economics.- 12. Chaos.- 12.1 What is Chaos?.- 12.2 The Lorenz Model. Motivation and Realization.- 12.3 How Chaos Occurs.- 12.4 Chaos and the Failure of the Slaving Principle.- 12.5 Correlation Function and Frequency Distribution.- 12.6 Discrete Maps, Period Doubling, Chaos, Intermittency.- 13. Some Historical Remarks and Outlook.- References, Further Reading, and Comments.