Time Reversibility, Computer Simulation, Algorithms, Chaos

Overview

Time Reversibility, Computer Simulation, Algorithms, Chaos

A small army of physicists, chemists, mathematicians, and engineers has joined forces to attack a classic problem, the "reversibility paradox", with modern tools. This book describes their work from the perspective of computer simulation, emphasizing the authors' approach to the problem of understanding the compatibility, and even inevitability, of the irreversible second law of thermodynamics with an underlying ...

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Overview

Time Reversibility, Computer Simulation, Algorithms, Chaos

A small army of physicists, chemists, mathematicians, and engineers has joined forces to attack a classic problem, the "reversibility paradox", with modern tools. This book describes their work from the perspective of computer simulation, emphasizing the authors' approach to the problem of understanding the compatibility, and even inevitability, of the irreversible second law of thermodynamics with an underlying time-reversible mechanics. Computer simulation has made it possible to probe reversibility from a variety of directions and "chaos theory" or "nonlinear dynamics" has supplied a useful vocabulary and a set of concepts, which allow a fuller explanation of irreversibility than that available to Boltzmann or to Green, Kubo and Onsager. Clear illustration of concepts is emphasized throughout, and reinforced with a glossary of technical terms from the specialized fields which have been combined here to focus on a common them.

The book begins with a discussion, contrasting the idealized reversibility of basic physics against the pragmatic irreversibility of real life. Computer models, and simulation, are next discussed and illustrated. Simulations provide the means to assimilate concepts through worked-out examples. State-of-the-art analyses, from the point of view of dynamical systems, are applied to many-body examples from non-equilibrium molecular dynamics and to chaotic irreversible flows from finite-difference, finite-element, and particle-based continuum simulations. Two necessary concepts from dynamical-systems theory - fractals and Lyapunov instability - are fundamental to the approach.

Undergraduate-level physics, calculus, and ordinary differential equations are sufficient background for a full appreciation of this book, which is intended for advanced undergraduates, graduates, and research workers. The generous assortment of examples worked out in the text will stimulate readers to explore the rich and fruitful field of study which links fundamental reversible laws of physics to the irreversibility surrounding us all.

This expanded edition stresses and illustrates computer algorithms with many new worked-out examples, and includes considerable new material on shockwaves, Lyapunov instability, and fluctuations.

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Product Details

  • ISBN-13: 9789814383165
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 6/28/2012
  • Edition number: 2
  • Pages: 401
  • Product dimensions: 6.00 (w) x 9.00 (h) x 0.80 (d)

Table of Contents

Preface vii

Preface to the First Edition xi

Glossary of Technical Terms xxi

1 Time Reversibility, Computer Simulation, Algorithms, Chaos 1

1.1 Microscopic Reversibility; Macroscopic Irreversibility 1

1.2 Time Reversibility of Irreversible Processes 6

1.3 Classical Microscopic and Macroscopic Simulation 8

1.4 Continuity, Information, and Bit Reversibility 10

1.5 Instability and Chaos 11

1.6 Simple Explanations of Complex Phenomena 13

1.7 The Paradox: Irreversibility from Reversible Dynamics 15

1.8 Algorithm: Fourth-Order Runge-Kutta Integrator 16

1.9 Example Problems 20

1.9.1 Equilibrium Baker Map 21

1.9.2 Equilibrium Galton Board 25

1.9.3 Equilibrium Hookean Pendulum 29

1.9.4 Nosé-Hoover Oscillator with a Temperature Gradient 32

1.10 Summary and Notes 36

1.10.1 Notes and References 37

2 Time-Reversibility in Physics and Computation 39

2.1 Introduction 39

2.2 Time Reversibility 41

2.3 Levesque and Verlet's Bit-Reversible Algorithm 44

2.4 Lagrangian and Hamiltonian Mechanics 46

2.5 Liouville's Incompressible Theorem 49

2.6 What is Macroscopic Thermodynamics? 50

2.7 First and Second Laws of Thermodynamics 52

2.8 Temperature, Zeroth Law, Reservoirs, Thermostats 54

2.9 Irreversibility from Stochastic Irreversible Equations 58

2.10 Irreversibility from Time-Reversible Equations? 60

2.11 An Algorithm Implementing Bit-Reversible Dynamics 61

2.12 Example Problems 67

2.12.1 Time-Reversible Dissipative Map 68

2.12.2 A Smooth-Potential Galton Board 73

2.13 Summary 77

2.13.1 Notes and References 78

3 Gibbs' Statistical Mechanics 81

3.1 Scope and History 81

3.2 Formal Structure of Gibbs' Statistical Mechanics 83

3.3 Initial Conditions, Boundary Conditions, Ergodicity 86

3.4 From Hamiltonian Dynamics to Gibbs' Probability 89

3.5 From Gibbs' Probability to Thermodynamics 90

3.6 Pressure and Energy from Gibbs' Canonical Ensemble 92

3.7 Gibbs' Entropy versus Boltzmann's Entropy 93

3.8 Number-Dependence and Thermodynamics Fluctuations 96

3.9 Green and Kubo's Linear-Response Theory 97

3.10 An Algorithm for Local Smooth-Particle Averages 99

3.11 Example Problems 103

3.11.1 Quasiharmonic Thermodynamics 104

3.11.2 Hard-Disk and Hard-Sphere Thermodynamics 106

3.11.3 Time-Reversible Confined Free Expansion 108

3.12 Summary 111

3.12.1 Notes and References 112

4 Irreversibility in Real Life 113

4.1 Introduction 113

4.2 Phenomenology - the Linear Dissipative Laws 116

4.3 Microscopic Basis of the Irreversible Linear Laws 117

4.4 Solving the Linear Macroscopic Equations 119

4.5 Nonequilibrium Entropy Changes 120

4.6 Fluctuations and Nonequilibrium States 123

4.7 Deviations from the Phenomenological Linear Laws 124

4.8 Causes of Irreversibility à la Boltzmann and Lyapunov 126

4.9 Rayleigh-Bénard Algorithm with Atomistic Flow 128

4.10 Rayleigh-Bénard Algorithm for a Continuum 135

4.11 Three Rayleigh-Bénard Example Problems 140

4.11.1 Rayleigh-Bénard Flow via Lorenz' Attractor 142

4.11.2 Rayleigh-Bénard Flow with Continuum Mechanics 144

4.11.3 Rayleigh-Bénard Flow with Molecular Dynamics 154

4.12 Summary 159

4.12.1 Notes and References 160

5 Microscopic Computer Simulation 163

5.1 Introduction 163

5.2 Integrating the Motion Equations 164

5.3 Interpretation of Results 165

5.4 Control of a Falling Particle 168

5.5 Second Law of Thermodynamics 176

5.6 Simulating Shear Flow and Heat Flow 177

5.7 Shockwaves 181

5.8 Algorithm for Periodic Shear Flow with Doll's Tensor 184

5.9 Example Problems 188

5.9.1 Isokinetic Nonequilibrium Galton Board 189

5.9.2 Heat-Conducting One-Dimensional Oscillator 192

5.9.3 Many-Body Heat Flow 195

5.10 Summary 196

5.10.1 Notes and References 197

6 Shockwaves Revisited 199

6.1 Introduction 199

6.2 Equation of State Information from Shockwaves 201

6.3 Shockwave Conditions for Molecular Dynamics 203

6.4 Shockwave Stability 206

6.5 Thermodynamic Variables 214

6.6 Shockwave Profiles from Continuum Mechanics 215

6.6.1 Shockwave Profile with Shear Viscosity 217

6.6.2 Shockwave Profile with Viscosity and Conductivity 220

6.6.3 Shockwave Profiles with Tensor Temperatures 222

6.6.4 Flow Algorithm with Maxwell-Cattaneo Time Delays 223

6.7 Comparing Model Profiles with Molecular Dynamics 229

6.8 Lyapunov Instability in Strong Shockwaves 232

6.9 Summary 238

6.9.1 Notes and References 238

7 Macroscopic Computer Simulation 241

7.1 Introduction 241

7.2 Continuity and Coordinate Systems 243

7.3 Macroscopic Flow Variables 245

7.4 Finite-Difference Methods 246

7.5 Finite-Element Methods 248

7.6 Smooth Particle Applied Mechanics [SPAM] 251

7.7 A SPAM Algorithm for Rayleigh-Bénard Convection 255

7.7.1 Initial Conditions 255

7.7.2 SPAM Evaluation of the Particle Densities 257

7.7.3 SPAM Evaluation of {∇u} and {∇T} 258

7.7.4 SPAM Evaluation of the Constitutive Relations 260

7.8 Applications of SPAM to Rayleigh-Bénard Flows 262

7.8.1 SPAM with and without a Core Potential 266

7.8.2 SPAM and Kinetic-Energy Fluctuations 268

7.9 Summary 271

7.9.1 Notes and References 271

8 Chaos, Lyapunov Instability, Fractals 273

8.1 Introduction 273

8.2 Continuum Mathematics 277

8.3 Chaos 278

8.4 The Spectrum of Lyapunov Exponents 279

8.5 Fractal Dimensions 284

8.6 A Simple Ergodic Fractal 288

8.7 Fractal Attractor-Repeller Pairs 290

8.8 A Global Second Law from Reversible Chaos 292

8.9 Coarse-Grained and Fine-Grained Entropy 297

8.10 Oscillators, Lyapunov Algorithms, Fractal Dimensions 298

8.10.1 A Thought-Provoking Oscillator Exercise 298

8.10.2 Doubly-Thermostated Oscillator; Lyapunov Spectra 300

8.10.3 Lyapunov Spectra from a Chaotic Double Pendulum 310

8.10.4 Coarse-Grained Galton Board Entropy 312

8.10.5 Color Conductivity 313

8.11 Summary 316

8.11.1 Notes and References 317

9 Resolving the Reversibility Paradox 319

9.1 Introduction 319

9.2 Irreversibility from Boltzmann's Kinetic Theory 320

9.3 Boltzmann's Equation Today 325

9.4 Gibbs' Statistical Mechanics 327

9.5 Jaynes' Information theory 330

9.6 Green and Kubo's Linear Response Theory 332

9.7 Thermomechanics 334

9.8 The Delay Times Separating Causes from their Effects 336

9.9 A Fluctuation Theorem 337

9.10 Are Initial Conditions Relevant? 340

9.11 Constrained Hamiltonian Ensembles 343

9.12 Anosov Systems and Sinai-Ruelle-Bowen Measures 344

9.13 Trajectories versus Distribution Functions 347

9.14 Are Maps Relevant? 348

9.15 Irreversibility ← Time-Reversible Motion Equations 351

9.16 Boltzmann-Equation Shockwave-Structure Algorithm 353

9.17 Summary 359

9.17.1 Notes and References 361

10 Afterword-a Research Perspective 363

10.1 Introduction 363

10.2 What do we Know? 364

10.3 Why Reversibility is Still a Problem 366

10.4 Change and Innovation 369

10.5 Role of Examples 372

10.6 Role of Chaos and Fractals 374

10.7 Role of Mathematics 374

10.8 Remaining Puzzles 376

10.9 Summary 379

10.10 Acknowledgments 383

Bibliography 387

Index 397

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