## Time Reversibility, Computer Simulation, and ChaosA 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 author's 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 set of concepts, which allow a fuller explanation of irreversibility than that available to Boltzmann or to Green and 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 theme.The book begins with a discussion contrasting the idealized reversibility of basic physics and 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 nonequilibrium 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. |

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### Contents

TimeReversibility in Physics and Computation | 29 |

Gibbs Statistical Mechanics | 61 |

Irreversibility in Real Life | 89 |

Microscopic Computer Simulation | 111 |

Macroscopic Computer Simulation | 141 |

Chaos Lyapunov Instability Fractals | 163 |

Resolving the Reversibility Paradox | 199 |

Afterworda Research Perspective | 229 |

Glossary of Technical Terms | 249 |

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### Common terms and phrases

algorithm analog Anosov systems approach averages Baker Map Boltzmann boundary conditions chaos chaotic coarse-grained collision comoving computer simulations continuum continuum mechanics convection coordinates corresponding derivatives described differential equations dissipative divergence ensemble entropy equations of motion equilibrium ergodic Eulerian example problems Figure fixed fluctuations fluid forces fractal friction coefficient function Galton Board Gibbs Hamiltonian heat conductivity heat flow heat flux heat reservoirs ideal-gas information dimension initial conditions isolated systems Lagrangian Law of Thermodynamics linear linear-response theory Liouville's Liouville's Theorem long-time-averaged Lyapunov exponents Lyapunov instability Lyapunov spectrum macroscopic many-body mass molecular dynamics momentum motion equations multifractal Newtonian nonequilibrium systems nonlinear Nose-Hoover numerical oscillator pairs particle perturbations phase-space physical probability density Rayleigh-Benard relative repellor reversibility Second Law Section shockwave simple simplest smooth-particle solution space stationary statistical mechanics strange attractor Theorem thermal thermomechanical thermostat thermostated time-averaged time-reversible tion trajectory transport coefficients two-dimensional typical understanding variables vector velocity viscosity