The publication of this second edition was motivated by several facts. First of all, the first edition had been sold out in less than one year. It had found excellent critics and enthusiastic responses from professors and students welcoming this new interdisciplinary approach. This appreciation is reflected by the fact that the book is presently translated into Russian and Japanese also. I have used this opportunity to include some of the most interesting recent developments. Therefore I have added a whole new chapter on the fascinating and rapidly growing field of chaos dealing with irregular motion caused by deterministic forces. This kind of phenomenon is presently found in quite diverse fields ranging from physics to biology. Furthermore I have included a section on the analytical treatment of a morphogenetic model using the order parameter concept developed in this book. Among the further additions, there is now a complete description of the onset of ultrashort laser pulses. It goes without· saying that the few minor mis prints or errors of the first edition have been corrected. I wish to thank all who have helped me to incorporate these additions.
Inhaltsverzeichnis
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. - 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 DynamicProcesses. - 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. 5 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. 1 l*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 Hierarchyof Laser Instabilities and Ultrashort Laser Pulses. - 8. 9 Instabilities in Fluid Dynamics: The Bénard and Taylor Problems. - 8. 10 The Basic Equations. - 8. 11 Damped and Neutral Solutions (R ? Rc). - 8. 12 Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations. - 8. 13 The Fokker-Planck Equation and Its Stationary Solution. - 8. 14 A Model for the Statistical Dynamics of the Gunn Instability Near Threshold. - 8. 15 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: A Stochastic Model for the Formation of Public Opinion. - 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 Further Examples of Chaotic Motion. - 13. Some Historical Remarks and Outlook. - References, Further Reading and Comments.
