Deterministic and Random Evolution

200 Pages·2013·1.675 MB·English

by  Jens Lorenz

MATHEMATICS RESEARCH DEVELOPMENTS

D R
ETERMINISTIC AND ANDOM
E
VOLUTION

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M R D
ATHEMATICS ESEARCH EVELOPMENTS

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MATHEMATICS RESEARCH DEVELOPMENTS

D R
ETERMINISTIC AND ANDOM
E
VOLUTION

JENS LORENZ
EDITOR

New York
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Contents
Preface ix
1 Introduction 1
2 Basic Concepts 5
1. Initial Value Problems for ODEs . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Discrete–Time Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3. Continuous vs. Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. Random Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Deterministic Systems:Outline of Advanced Topics 13
1. Planetary Motion: Example for Determinism . . . . . . . . . . . . . . . . . 13
2. Reversibility in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. Sensitive Dependence on Initial Conditions . . . . . . . . . . . . . . . . . . 14
4. Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5. Dependence on Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6. Variation on Different Time Scales . . . . . . . . . . . . . . . . . . . . . . . 17
4 Planetary Motion 21
1. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2. The Two Body Problem: Reduction to One Bodyin a Central Field . . . . . 22
3. One Body in a Central Field . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4. The Equation for an Ellipse in Polar Coordinates . . . . . . . . . . . . . . . 26
5. The Kepler Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6. Kepler’s Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7. Time Dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8. Bessel Functions via a Generating Function:Integral Representation. . . . . 37
9. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
vi Jens Lorenz
5 Is Time Reversible? 41
1. Reversibility for the Two Body Problem . . . . . . . . . . . . . . . . . . . . 41
2. Reversibility: General Definition . . . . . . . . . . . . . . . . . . . . . . . . 42
3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 The Bernoulli Shift and the Logistic Map 45
1. The Bernoulli Shift: Definition . . . . . . . . . . . . . . . . . . . . . . . . . 45
2. The Bernoulli Shift: Dynamical Properties . . . . . . . . . . . . . . . . . . . 47
3. The Logistic Map and Its Relation tothe Bernoulli Shift . . . . . . . . . . . 49
4. Average Behavior of the Logistic Map . . . . . . . . . . . . . . . . . . . . . 50
7 Evolution on Two Time–Scales 55
1. Fast and Slow Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2. Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3. A Difficult Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Stability and Bifurcations 61
1. Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2. Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3. Logistic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4. The Delayed Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5. Parameter Dependent Evolution and Hysteresis . . . . . . . . . . . . . . . . 70
9 Scripts 77
1. Script for Logistic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2. Scripts for the Delayed Logistic Map . . . . . . . . . . . . . . . . . . . . . . 78
3. Scripts for Parameter Dependent Evolutionand Hysteresis . . . . . . . . . . 79
10 Two Oscillators: Periodicity,Ergodicity, and Phase Locking 83
1. The Circle and the Two–Torus . . . . . . . . . . . . . . . . . . . . . . . . . 83
2. Uncoupled Oscillators: Periodic Solutions . . . . . . . . . . . . . . . . . . . 85
3. Uncoupled Oscillators: Ergodicity. . . . . . . . . . . . . . . . . . . . . . . . 88
4. Time Average Equals Space Average for a Circle Map . . . . . . . . . . . . 89
4.1. A Poincar´e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2. Ergodicity of a Circle Map . . . . . . . . . . . . . . . . . . . . . . . 90
5. Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11 The Gambler’s Ruin Problem 101
1. Description of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2. Some Questions and Numerical Realization . . . . . . . . . . . . . . . . . . 102
3. The Transition Matrix P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4. Evolution of Probability Density . . . . . . . . . . . . . . . . . . . . . . . . 106
Contents vii
5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7. Script: Evolving the Probability Densityfor Gambler’s Ruin . . . . . . . . . 109
12 Gambler’s Ruin: Probabilities and Expected Time 111
1. Probability of Ruin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2. Probability of Winning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3. Expected Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4. The Matrix View: Limit of Probability Densities . . . . . . . . . . . . . . . 115
4.1. The Eigenvalues of P . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2. Convergence of Probability Density Vectors . . . . . . . . . . . . . . 119
13 Stochastic Model of a Simple Growth Process 121
1. Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2. The Forward Kolmogorov Equations . . . . . . . . . . . . . . . . . . . . . . 122
3. Solution of the Forward Kolmogorov Equations . . . . . . . . . . . . . . . . 124
4. The Sum of the p (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
j
5. The Expected Value of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
t
5.1. The Expected Value of a Discrete Random Variable . . . . . . . . . 130
5.2. The Expected Value E(X ) . . . . . . . . . . . . . . . . . . . . . . . 131
t
6. The Variance of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
t
6.1. The Variance of a Discrete Random Variable . . . . . . . . . . . . . 134
6.2. The Variance Var(X ) . . . . . . . . . . . . . . . . . . . . . . . . . . 135
t
7. Statistics of Interevent Times . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1. Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2. Statistical Law for Interevent Times . . . . . . . . . . . . . . . . . . 139
8. Numerical Realization of Random Evolution . . . . . . . . . . . . . . . . . . 141
9. Figures and Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
14 Introduction to Kinetic Theory 149
1. Boyle, Bernoulli, Maxwell, and Sadi Carnot . . . . . . . . . . . . . . . . . . 149
2. Daniel Bernoulli: Pressure, Volume,and Particle Velocity . . . . . . . . . . . 151
2.1. The Ideal Gas Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3. The Maxwell–Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . 154
3.1. Assumptions Leading to Maxwell’s Distribution . . . . . . . . . . . . 156
3.2. Derivation of Maxwell’s Distribution . . . . . . . . . . . . . . . . . . 156
3.3. Relation between K and T . . . . . . . . . . . . . . . . . . . . . . . 158
4. Optimal Heat Engines: Carnot’s Cycle . . . . . . . . . . . . . . . . . . . . . 160
4.1. Heat Energy and Mechanical Energy . . . . . . . . . . . . . . . . . . 160
4.2. Carnot’s Heat Engine: Preparations . . . . . . . . . . . . . . . . . . 161
4.3. The Carnot Cycle and the Direction of Time . . . . . . . . . . . . . 165
viii Jens Lorenz
15 Random Evolution: From Order to Chaos 171
1. Random Evolution of Macro States . . . . . . . . . . . . . . . . . . . . . . . 171
2. Macro States and Micro States . . . . . . . . . . . . . . . . . . . . . . . . . 175
3. Back to the Ordered State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
References
Index 185