A recent development is the discovery that simple systems of equations can have chaotic solutions in which small changes in initial conditions have a large effect on the outcome, rendering the corresponding experiments effectively irreproducible and unpredictable. An earlier book in this sequence, Elegant Chaos: Algebraically Simple Chaotic Flows provided several hundred examples of such systems, nearly all of which are purely mathematical without any obvious connection with actual physical processes and with very limited discussion and analysis. In this book, we focus on a much smaller subset of such models, chosen because they simulate some common or important physical phenomenon, usually involving the motion of a limited number of point-like particles, and we discuss these models in much greater detail. As with the earlier book, the chosen models are the mathematically simplest formulations that exhibit the phenomena of interest, and thus they are what we consider 'elegant.' Elegant models, stripped of unnecessary detail while maximizing clarity, beauty, and simplicity, occupy common ground bordering both real-world modeling and aesthetic mathematical analyses. A computational search led one of us (JCS) to the same set of differential equations previously used by the other (WGH) to connect the classical dynamics of Newton and Hamilton to macroscopic thermodynamics. This joint book displays and explores dozens of such relatively simple models meeting the criteria of elegance, taste, and beauty in structure, style, and consequence. This book should be of interest to students and researchers who enjoy simulating and studying complex particle motions with unusual dynamical behaviors. The book assumes only an elementary knowledge of calculus. The systems are initial-value iterated maps and ordinary differential equations but they must be solved numerically. Thus for readers a formal differential equations course is not at all necessary, of little value and limited use.
Author(s): Julien Clinton Sprott, William Graham Hoover, Carol Griswold Hoover
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 324
City: Singapore
Contents
Preface
1. Linear Oscillators
1.1 Simple Harmonic Oscillator
1.2 Damped Harmonic Oscillator
1.2.1 Overdamped case
1.2.2 Critically damped case
1.2.3 Underdamped case
1.2.4 Undamped case
1.2.5 Antidamped oscillations
1.2.6 Critical antidamping
1.2.7 Extreme antidamping
1.3 Periodically Forced Harmonic Oscillator
1.3.1 Damped case
1.3.2 Undamped case
1.4 Two Coupled Harmonic Oscillators
1.4.1 Moderate coupling
1.4.2 Weak coupling
1.4.3 Strong coupling
1.5 Harmonic Oscillator Chains
1.5.1 Three coupled oscillators
1.5.2 Long chain of oscillators
1.5.3 Ring of oscillators
1.6 Primer on Linear Algebra
1.6.1 Calculation of eigenvalues and eigenvectors
1.6.2 Saddle points
2. Nonlinear Oscillators
2.1 Simple Pendulum
2.2 Damped Pendulum
2.3 Periodically Forced Pendulum
2.3.1 Undamped case
2.3.2 Lyapunov exponents
2.3.3 Damped case
2.3.4 Kaplan–Yorke dimension
2.4 Duffing Oscillator
2.4.1 Softening spring
2.4.2 Hardening spring
2.4.3 Quartic potential
2.4.4 Two-well potential
2.5 Forced Square-Well Oscillator
2.5.1 Damped case
2.5.2 Undamped case
2.6 Asymmetric-Well Oscillator
2.7 Nonlinearly Damped Harmonic Oscillator
2.8 van der Pol Oscillator
2.8.1 Unforced case
2.8.2 Periodically forced case
2.9 Periodically Damped Oscillator
2.9.1 Unforced case
2.9.2 Periodically forced case
2.10 Rayleigh Oscillator
2.11 Rayleigh–Duffing Two-Well Oscillator
2.11.1 Unforced case
2.11.2 Periodically forced case
2.12 Parametrically Forced Pendulum
2.13 Non-Deterministic Harmonic Oscillator
3. Coupled Oscillators
3.1 Coupled Quartic Oscillators
3.1.1 Undamped case
3.1.2 Damped case
3.2 Coupled Pendulums
3.2.1 Undamped case
3.2.2 Damped case
3.3 Master–Slave Oscillators
3.3.1 Undamped case
3.3.2 Damped case
3.3.3 Simplified case
3.4 Coupled van der Pol Oscillators
3.4.1 Symmetric case
3.4.2 Simplified case
3.4.3 Master–slave case
3.4.4 Parametrically coupled case
3.5 Ball on an Oscillating Floor
3.6 Nonlinearly Coupled Harmonic Oscillators
3.7 Lotka–Volterra Systems
4. Thermostatted Oscillators
4.1 Nosé–Hoover Oscillator
4.1.1 Conservative Nosé–Hoover oscillator
4.1.2 Dissipative Nosé–Hoover oscillator
4.1.3 Nosé–Hoover with an unstable thermostat
4.2 Cubic Thermostat Oscillator
4.3 Chain Thermostat Oscillators
4.3.1 Martyna–Klein–Tuckerman oscillator
4.3.2 Hoover–Holian oscillator
4.3.3 Ju–Bulgac oscillator
4.4 Buncha Oscillator
4.5 Logistic Thermostat Oscillator
4.6 Signum Thermostatted Linear Oscillator
4.7 Signum Thermostatted Nonlinear Oscillators
4.7.1 Ergodic cubic oscillator
4.7.2 Ergodic Duffing oscillator
4.7.3 Ergodic pendulum
4.7.4 Square-well oscillator
4.8 Dissipative Signum Thermostat
5. Two-Dimensional Oscillators
5.1 Linear Oscillators
5.1.1 Isotropic oscillator
5.1.2 Anisotropic oscillator
5.1.3 Periodically forced oscillator
5.2 Nonlinear Oscillators
5.2.1 Hardening springs
5.2.1.1 Conservative case
5.2.1.2 Dissipative case
5.2.2 Mexican hat potential
5.2.3 Springy pendulum
5.2.4 Diatomic molecule
5.2.5 Hénon–Heiles system
5.2.6 Particle in periodic potential
5.3 Thermostatted Oscillators
5.3.1 Two-dimensional Nosé–Hoover oscillator
5.3.2 Two-dimensional nonlinear oscillator
5.3.3 Two-dimensional signum thermostat oscillator
5.4 Chaotic Scattering
5.4.1 Bunimovich stadium
5.4.2 Lorentz gas
5.4.3 Particle in cell
5.4.4 Galton board
5.4.5 Fermi–Ulam model
6. Map and Walk Analogs of Flows
6.1 Maps as Analogs of Flows
6.2 Chaos and Ergodicity in One Dimension
6.3 Time-Reversible Conservative Maps
6.4 Time-Reversible Nonequilibrium Maps
6.5 Fractal Information Dimensions
6.6 Mesh Dependence of Information Dimension
6.7 Random Walk Equivalents of Maps
6.8 Further Fractal Time-Reversible Maps
7. From Small Systems to Large
7.1 Bridging the Gap between Small and Large Systems
7.2 Equilibrium Systems with Different Scales
7.3 Collisionless Knudsen Gas Boundary Conditions
7.4 Hamilton's Equations; Coordinates and Momenta
7.5 Feedback Control of Atomistic Simulations
7.6 The Nosé and Nosé–Hoover Oscillators
7.7 Hamilton's Motion Equations; Kinetic Temperature
7.8 Many-Body Simulations - Repulsive Pairwise Forces
7.9 A Smooth Finite-Range Soft-Disk Potential
7.10 Energy and Pressure for Isothermal Soft Disks
7.11 Representations of Equation of State Data
7.12 Lindemann Criterion for Melting
7.13 Centered Second Difference Newtonian Integration
7.14 Fourth-Order Classic Runge–Kutta Integration
8. Thermodynamics and Molecular Dynamics
8.1 Macroscopic Thermodynamics: Heat, Work, Energy
8.2 A State Function Associated with Heat, Entropy
8.3 Thermodynamic Entropy from Carnot's 1824 Cycle
8.4 Kinetic Theory and the Boltzmann Equation
8.5 van der Waals' 1873 Model for Liquids and Gases
8.6 Sub-Spinodal Evolution with Lennard-Jones' Potential
8.7 Boltzmann and Gibbs' Statistical Mechanics
8.8 Liouville's Theorem and Gibbs' Ensembles
8.9 Entropy in Statistical Mechanics
8.10 Entropies from Phase Space Microstates
8.11 From the Microcanonical to the Canonical Ensemble
8.12 Nosé–Hoover and Hoover-Holian Moments
8.13 From the Virial Theorem to the Pressure Tensor
8.14 Gravitational Equilibria with Molecular Dynamics
8.15 Isoenergetic Applications of Thermodynamics
8.16 An Application of the Second Law of Thermodynamics
9. Mechanics of Nonequilibrium Fluids
9.1 Nonequilibrium Systems
9.2 The Continuum View of Nonequilibrium Flows
9.3 The Navier–Stokes Equations
9.4 Steady-State Shear Viscosity for Soft Disks
9.5 Shear Viscosity Simulations using Doll's Tensor
9.6 Heat Conduction with a One-Dimensional Model
9.7 Alternative Thermostats
9.8 Navier–Stokes Shock Wave Structure
10. Micro and Macro Time-Reversibility
10.1 Microscopic and Macroscopic Time-Reversibility
10.2 Time-Reversible Centered Second Differences
10.3 Loschmidt's and Zermélo's Paradoxes
10.4 One-Dimensional Conducting Oscillator
10.5 Conducting Doubly Thermostatted Oscillator
10.6 Resolution of the Paradoxes
10.7 Smooth-Particle Averaging for Field Variables
10.8 Nonequilibrium Simulations
10.9 Newtonian Simulations of Shock Wave Structure
10.10 Tensorial Structure of the Steady Shock Wave
10.11 Additional Points Along the Shock Hugoniot Curve
10.12 One-Dimensional Planar Shock Waves are Stable
10.13 Rarefaction from Reversed Irreversible Shock Waves
10.14 Melting and Freezing for Hard Disks and Spheres
11. Attractions in Molecular Dynamics
11.1 Attractive Forces Produce Condensed Matter
11.2 Alternatives to Lennard-Jones' Potential
11.3 Initial Conditions for Liquid Phase Simulations
11.4 Inelastic Two-Ball Collisions with Attractive Forces
11.5 Irreversibility of the Reversed Two-Ball Problems
11.6 The Reversal of Irreversible Processes
11.7 Irreversibility, Restitution, and the One-Ball Problem
11.8 Interesting Equilibria and Research Ideas
11.9 Smooth-Particle Approach to Liquid Problems
11.10 Parting Comments
Bibliography
Index
About the Authors
