Handbook of Dynamics and Probability

Description

Our time is characterized by an explosive growth in the use of ever more complicated and sophisticated (computer) models. These models rely on dynamical systems theory for the interpretation of their results and on probability theory for the quantification of their uncertainties. A conscientious and intelligent use of these models requires that both these theories are properly understood. This book is to provide such understanding. It gives a unifying treatment of dynamical systems theory and probability theory. It covers the basic concepts and statements of these theories, their interrelations, and their applications to scientific reasoning and physics. The book stresses the underlying concepts and mathematical structures but is written in a simple and illuminating manner without sacrificing too much mathematical rigor. The book is aimed at students, post-docs, and researchers in the applied sciences who aspire to better understand the conceptual and mathematical underpinnings of the models that they use. Despite the peculiarities of any applied science, dynamics and probability are the common and indispensable tools in any modeling effort. The book is self-contained, with many technical aspects covered in appendices, but does require some basic knowledge in analysis, linear algebra, and physics. Peter Müller, now a professor emeritus at the University of Hawaii, has worked extensively on ocean and climate models and the foundations of complex system theories. Foreword Preface Contents 1 Introduction Part I Dynamics 2 Deterministic Dynamical Systems 2.1 Introduction 2.2 States and Their Time Evolution 2.3 Discrete Systems and Iterative Maps 2.4 Continuous Systems and ODEs 2.5 Continuous Systems and PDEs 2.6 Observables and Parameters 2.7 Invariant Sets and Orbits 2.8 Summary References 3 Linear and Integrable Systems 3.1 Introduction 3.2 Vector Spaces 3.3 Linear Dynamical Systems 3.4 Eigenanalysis 3.5 Nonlinear Interactions 3.6 Integrable Systems 3.7 The KAM Theorem 3.8 Dimensions of Dynamical Systems 3.9 Summary References 4 Stability and Long-Time Behavior 4.1 Introduction 4.2 Topology 4.3 Stability 4.4 Linear Stability of Fixed Points 4.5 Liapunov Function 4.6 Poincaré Maps 4.7 Linear Stability of Periodic Orbits 4.8 Sensitivity to Initial Conditions 4.9 Limiting Sets and Attractors 4.10 State Space Portrait 4.11 Ingredients of Chaos 4.12 Bifurcations 4.13 Summary References 5 Ergodic Theory 5.1 Introduction 5.2 The Evolution of Volumes 5.3 Measure Spaces 5.4 Invariant Measure 5.5 Poincaré Recurrence Theorem 5.6 Time Averages 5.7 Ergodic Measures 5.8 Ergodic System 5.9 Mixing and Exact Systems 5.10 Shift Map 5.11 Summary References 6 Numerical Algorithms 6.1 Introduction 6.2 Round-Off Error 6.3 Truncation Errors 6.4 Stability 6.5 Numerical Diffusion and Dispersion 6.6 Conservation Laws 6.7 Spectral and Finite Element Models 6.8 Summary Reference Part II Probability 7 Probability Theory 7.1 Introduction 7.2 Probability Spaces 7.3 Conditional Probabilities and Statistical Independence 7.4 Random Variables and Their Distributions 7.5 Expectation Values 7.6 Moments 7.7 Bi- and Multivariate Random Variables 7.8 Some Probability Distributions 7.9 Convergence of Sequences of Random Variables 7.10 Weak and Strong Law of Large Numbers 7.11 Central Limit Theorem 7.12 Extreme Value Theory 7.13 Summary Reference 8 Discrete-Time Stochastic Processes 8.1 Introduction 8.2 The Canonical Probability Space 8.3 Discrete Stochastic Processes 8.4 Independent Chains 8.5 Markov Chains 8.6 The Markov Operator 8.7 Asymptotic Behavior 8.8 Random Walks 8.9 Summary 9 Continuous-Time Stochastic Processes 9.1 Introduction 9.2 Separable Modifications 9.3 Markov Processes 9.4 Master Equation 9.5 Fokker-Planck Equation 9.6 The Wiener Process 9.7 Gaussian White Noise 9.8 Summary References 10 Information Entropy 10.1 Introduction 10.2 Information Content 10.3 Entropy of a Discrete Random Variable 10.4 Entropy of Bi- and Multivariate Random Variables 10.5 Asymptotic Equipartition Theorem 10.6 Entropy of Continuous Random Variables 10.7 Coarse-Graining and Smoothing 10.8 The Principle of Maximum Entropy 10.9 Summary References Part III Probability in Dynamical Evolution 11 Time Evolution of Broadened Initial States 11.1 Introduction 11.2 Broadened Versus Pure Initial States 11.3 Frobenius–Perron Operator 11.4 Liouville Equation 11.5 Stationary Densities 11.6 Convergence Toward Stationary Densities 11.7 Behavior of the Entropy 11.8 Summary References 12 Stochastic Processes Generated by Observables 12.1 Introduction 12.2 Relation Between Stochastic Processes and Dynamical Systems 12.3 Coarse-Graining 12.4 Elimination of Variables 12.5 Adiabatic Elimination of Fast Variables 12.6 Summary Reference 13 Stochastic Dynamical Systems 13.1 Introduction 13.2 Autoregressive Processes 13.3 Multiplicative Processes 13.4 Langevin Equation 13.5 Stochastic Differential Equations 13.6 Summary References Part IV Probability in Scientific Reasoning 14 Interpretations of Probability 14.1 Introduction 14.2 The Physicalist's Interpretation 14.3 The Frequentist's Interpretation 14.4 The Logicist's Interpretation 14.5 The Subjectivist's Interpretation 14.6 The Utilitarian Approach 14.7 Summary References 15 Parameter Estimation and Hypothesis Testing 15.1 Introduction 15.2 Sampling 15.3 Estimator Versus Estimate 15.4 Interpretation of Estimates 15.5 Maximum Likelihood Estimators 15.6 Mean Square Error 15.7 Minimizing Risk 15.8 Hypothesis Testing 15.9 Type I and II Errors 15.10 Likelihood Ratio Tests 15.11 Summary 16 Bayesian Inferences 16.1 Introduction 16.2 Bayes Theorem 16.3 The Bayesian Paradigm 16.4 Bayesian Estimators 16.5 Bayesian Hypothesis Testing 16.6 Comparison of Frequentist and Bayesian Approach 16.7 Choice of Prior 16.8 Statistical Inference 16.9 Decisions 16.10 Data Assimilation 16.11 Summary References Part V Probability in Physics 17 Equilibrium Statistical Mechanics 17.1 Introduction 17.2 Thermostatics 17.3 Implications of PME 17.4 The Micro-canonical Ensemble 17.5 Quantum-Mechanical Corrections 17.6 The Canonical Ensemble 17.7 Macro-Variables 17.8 The Thermodynamic Limit 17.9 The Ideal Gas of Particles 17.10 Summary References 18 Non-equilibrium Statistical Mechanics 18.1 Introduction 18.2 Fluid Dynamics 18.3 The Continuum Limit of a N-particle System 18.4 Average of a N-particle System 18.5 Applying PME 18.6 Summary References 19 Foundational Issues of Statistical Mechanics 19.1 Introduction 19.2 The Arrow of Time Versus Time-Reversible Systems 19.3 Entropy 19.4 Dynamical Constraints 19.5 Irreversibility and Microscopic Dynamics 19.6 Ensemble 19.7 Summary References 20 Quantum Mechanics 20.1 Introduction 20.2 The Pragmatic Rules 20.3 Pure States 20.4 Mixed States 20.5 Entangled States 20.6 Quantum Probability 20.7 Classical Versus Quantum Logic 20.8 The Foundational Issues 20.9 Summary References Appendix A Classical Mechanics A.1 Newtonian dynamics A.2 Hamilton's principle A.3 Hamiltonian Dynamics A.4 Harmonic Oscillator Appendix B Thermostatics B.1 Thermodynamic Variables and Processes B.2 First Law of Thermodynamics B.3 Second Law of Thermodynamics B.4 Thermodynamic Potentials B.5 Thermodynamic Limit Appendix C Fluid Dynamics C.1 Irreversible Fluid Dynamics C.2 Navier–Stokes and Euler Equations C.3 Eulerian Versus Lagrangian Description C.4 Variational Principles for Perfect Fluids Appendix D Some Proofs and Explicit Calculations D.1 Derivation of the Lorenz Equations D.2 Inverse Scattering Transform of the KdV Equation D.3 Adiabatic Elimination of Fast Variables D.4 Balance Equations D.5 Reduced Densities Appendix E Mathematical Tools, Conventions, and Notation E.1 Sets E.2 Functions E.3 Groups E.4 Vector Spaces E.5 Normed Spaces E.6 Metric Spaces E.7 Limits E.8 Linear Operators E.9 Matrix Notation E.10 Dirac Notation E.11 Eigenvectors and Eigenvalues E.12 Topological Sets E.13 Measure Spaces E.14 Two Important Inequalities E.15 Permutations and Combinations E.16 Calculus of Variation E.17 Legendre Transformation E.18 Classical Propositional Logic E.19 Fourier Transforms and Series E.20 Jacobian E.21 Stirling's Formula References Index