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Tables of Contents for Time Reversibility, Computer Simulation, and Chaos
Chapter/Section Title
Page #
Page Count
Frontispiece and Dedication
v
Preface
vii
Time Reversibility, Computer Simulation, Chaos
1
28
Microscopic Time Reversibility and Macroscopic Irreversibility
1
5
Time-Reversible Theories of Irreversible Processes
6
2
Classical Microscopic and Macroscopic Simulation
8
1
Continuity, Information, and Bit Reversibility
9
2
Instability and Chaos
11
2
Simple Explanations of Complex Phenomena
13
2
Reversibility Paradox: Irreversibility from Reversible Dynamics
15
1
Example Problems
16
11
Equilibrium Baker Map
16
5
Equilibrium Galton Board
21
4
Equilibrium Hookean Pendulum
25
2
Summary
27
2
Time-Reversibility in Physics and Computation
29
32
Introduction
29
2
Time Reversibility
31
2
Levesque and Verlet's Bit-Reversible Algorithm
33
3
Lagrangian and Hamiltonian Mechanics
36
2
Liouville's Incompressible Theorem
38
1
What is Macroscopic Thermodynamics?
39
2
First and Second Laws of Thermodynamics
41
2
Temperature, Zeroth Law, Reservoirs, and Thermostats
43
4
Irreversibility from Stochastic Irreversible Equations
47
3
Irreversibility from Time-Reversible Equations?
50
1
Example Problems
51
7
Time-Reversible Dissipative Map
52
5
Time-Reversible Smooth Galton Board
57
1
Summary
58
3
Gibbs' Statistical Mechanics
61
28
Introduction
61
2
Formal Structure of Statistical Mechanics
63
3
Initial Conditions, Boundary Conditions, Ergodicity
66
3
From Hamiltonian Dynamics to Gibbs' Probability
69
2
From Gibbs' Probability to Thermodynamics
71
2
Pressure and Energy from Gibbs' Canonical Ensemble
73
1
Gibbs' Entropy versus Boltzmann's Entropy
74
3
Number-Dependence and Thermodynamic Fluctuations
77
1
Green and Kubo's Linear-Response Theory
77
2
Example Problems
79
8
Quasiharmonic Thermodynamics
80
2
Hard-Sphere Thermodynamics
82
2
Time-Reversible Confined Free Expansion
84
3
Summary
87
2
Irreversibility in Real Life
89
22
Introduction
89
3
The Phenomenological Linear Laws
92
1
Microscopic Basis of Linear Irreversibility
93
2
Solving the Linear Macroscopic Equations
95
1
Nonequilibrium Entropy Changes
96
3
Fluctuations and Nonequilibrium States
99
2
Deviations from the Phenomenological Linear Laws
101
1
Causes of Irreversibility a la Boltzmann and Lyapunov
102
2
Example Problems
104
5
Rayleigh-Benard Flow via Lorenz' Attractor
105
1
Rayleigh-Benard Flow with Atoms
106
3
Summary
109
2
Microscopic Computer Simulation
111
30
Introduction
111
1
Integrating the Motion Equations
112
1
Interpretation of Results
113
2
Control of a Falling Particle
115
3
Liouville's Theorems and Nonequilibrium Stability
118
5
Second Law of Thermodynamics
123
1
Simulating Shear Flow and Heat Flow
123
5
Shockwaves
128
2
Example Problems
130
9
Isokinetic Nonequilibrium Galton Board
130
4
Heat-Conducting One-Dimensional Oscillator
134
3
Many-Body Heat Flow
137
2
Summary
139
2
Macroscopic Computer Simulation
141
22
Introduction
141
2
Continuity and Coordinate Systems
143
2
Macroscopic Flow Variables
145
1
Finite-Difference Methods
146
2
Finite-Element Methods
148
2
Smooth Particle Applied Mechanics
150
3
Example Problems
153
8
Rayleigh-Benard Flow with Finite Differences
154
4
Rayleigh-Benard Flow with Smooth Particles
158
3
Summary
161
2
Chaos, Lyapunov Instability, Fractals
163
36
Introduction
163
4
Continuum Mathematics
167
1
Chaos
168
1
Spectrum of Lyapunov Exponents
169
6
Fractal Dimensions
175
3
A Simple Ergodic Fractal
178
2
Fractal Attractor-Repellor Pairs
180
2
A Global Second Law from Reversible Chaos
182
5
Coarse-Grained and Fine-Grained Entropy
187
2
Example Problems
189
9
Chaotic Double Pendulum
190
1
Coarse-Grained Galton Board Entropy
191
2
Heat-Conducting Harmonic Oscillator
193
1
Color Conductivity
194
4
Summary
198
1
Resolving the Reversibility Paradox
199
30
Introduction
199
1
Irreversibility from Boltzmann's Kinetic Theory
200
5
Boltzmann's Equation Today
205
2
Gibbs' Statistical Mechanics
207
3
Jaynes' Information Theory
210
1
Green and Kubo's Linear Response Theory
211
2
Thermomechanics
213
2
Are Initial Conditions Relevant?
215
3
Constrained Hamiltonian Ensembles
218
1
Anosov Systems and Sinai-Ruelle-Bowen Measures
219
3
Trajectories versus Distribution Functions
222
1
Are Maps Relevant?
223
1
Irreversibility from Time-Reversible Motion Equations
224
2
Summary
226
3
Afterword---a Research Perspective
229
20
Introduction
229
1
What do we know?
230
2
Why Reversibility is Still a Problem
232
3
Change and Innovation
235
3
Role of Examples
238
2
Role of Chaos and Fractals
240
1
Role of Mathematics
240
2
Remaining Puzzles
242
3
Summary
245
2
Acknowledgments
247
2
Glossary of Technical Terms
249
4
Alphabetical Bibliography
253
6
Index
259