Preface v

Chapter 1. Preliminaries 1

1.1. Least Squares Approximation 1

1.2. Orthonormal Bases 7

1.3. Fourier Series 9

1.4. Fourier Transform 12

1.5. Exercises 16

1.6. Bibliography 16

Chapter 2. Probability 17

2.1. Definitions 17

2.2. Expected Values and Moments 20

2.3. Monte Carlo Methods 26

2.4. Parametric Estimation 29

2.5. The Central Limit Theorem 31

2.6. Conditional Probability and Conditional Expectation 34

2.7. Bayes’ Theorem 38

2.8. Exercises 40

2.9. Bibliography 42

Chapter 3. Brownian Motion 43

3.1. Definition of Brownian Motion 43

3.2. Brownian Motion and the Heat Equation 45

3.3. Solution of the Heat Equation by Random Walks 47

3.4. The Wiener Measure 50

3.5. Heat Equation with Potential 52

3.6. Physicists’ Notation for Wiener Measure 55

3.7. Another Connection Between Brownian Motion and the

Heat Equation 57

3.8. First Discussion of the Langevin Equation 59

3.9. Solution of a Nonlinear Differential Equation by Branching

Brownian Motion 64

3.10. A Brief Introduction to Stochastic ODEs 65

3.11. Exercises 67

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viii CONTENTS

3.12. Bibliography 69

Chapter 4. Stationary Stochastic Processes 71

4.1. Weak Definition of a Stochastic Process 71

4.2. Covariance and Spectrum 74

4.3. The Inertial Spectrum of Turbulence 76

4.4. Random Measures and Random Fourier Transforms 78

4.5. Prediction for Stationary Stochastic Processes 84

4.6. Data Assimilation 89

4.7. Exercises 92

4.8. Bibliography 95

Chapter 5. Statistical Mechanics 97

5.1. Mechanics 97

5.2. Statistical Mechanics 99

5.3. Entropy and Equilibrium 102

5.4. The Ising Model 105

5.5. Markov Chain Monte Carlo 107

5.6. Renormalization 111

5.7. Exercises 116

5.8. Bibliography 117

Chapter 6. Time-Dependent Statistical Mechanics 119

6.1. More on the Langevin Equation 119

6.2. A Coupled System of Harmonic Oscillators 122

6.3. Mathematical Addenda 124

6.4. The Mori-Zwanzig Formalism 129

6.5. More on Fluctuation-Dissipation Theorems 134

6.6. Scale Separation and Weak Coupling 135

6.7. Noninstantaneous Memory 137

6.8. Exercises 141

6.9. Bibliography 142

Index 145