Computational Physics

Contains a wealth of topics to allow instructors flexibility in the choice of topics and depth of coverage: Examines projective motion with and without realistic air resistance. Discusses planetary motion and the three-body problem. Explores chaotic motion of the pendulum and waves on a string. Includes topics relating to fractal growth and stochastic systems. Offers examples on statistical physics and quantum mechanics. Contains ample explanations of the necessary algorithms students need to help them write original programs, and provides many example programs and calculations for reference.

Nicholas Giordano obtained his B.S. at Purdue University and his Ph.D. at Yale University. He has been on the faculty at Purdue since 1979, served as an Assistant Dean of Science from 2000-2003, and is currently the Hubert James Distinguished Professor of Physics. His research interests include electrical conduction, superconductivity, and magnetism in ultra-small metallic structures, along with musical acoustics and the physics of the piano. Ideas for this book grew out of the course on computational physics that he developed and taught in the early 1990s. Professor Giordano earned a Computational Science Education Award from the Department of Energy in 1997, and in 2004 was named Indiana Professor of the Year by the Carnegie Foundation for the Advancement of Teaching and the Council for the Advancement and Support of Education.

Hisao Nakanishi earned his B.S. from Brown University and his Ph.D. from Harvard University. His Ph.D. research concerned scaling and universality in a geometric phase transition called percolation and he has been interested in scale-invariance ever since. During his first postdoctoral work at Cornell he was introduced to the problem of surface critical phenomena such as wetting phase transitions, and later at the University of California, Santa Barbara, he started working on the statistics of diffusion and polymers in earnest. .In 1992 Professor Nakanishi was a part of the team that won a Gordon Bell Prize for the application of parallel computing to a problem in polymer statistics. More recently he has also put on another hat as a developer of a computer-based interactive exercise system which is used by a few thousand students at Purdue each year.

Preface

ix

About the Authors

xii

A First Numerical Problem

1

(17)

Radioactive Decay

1

(1)

A Numerical Approach

2

(1)

Design and Construction of a Working Program: Codes and Pseudocodes

3

(8)

Testing Your Program

11

(1)

Numerical Considerations

12

(2)

Programming Guidelines and Philosophy

14

(4)

Realistic Projectile Motion

18

(30)

Bicycle Racing: The Effect of Air Resistance

18

(7)

Projectile Motion: The Trajectory of a Cannon Shell

25

(6)

Baseball: Motion of a Batted Ball

31

(5)

Throwing a Baseball: The Effects of Spin

36

(8)

Golf

44

(4)

Oscillatory Motion and Chaos

48

(46)

Simple Harmonic Motion

48

(6)

Making the Pendulum More Interesting: Adding Dissipation, Non-linearity, and a Driving Force

54

(4)

Chaos in the Driven Nonlinear Pendulum

58

(8)

Routes to Chaos: Period Doubling

66

(4)

The Logistic Map: Why the Period Doubles

70

(5)

The Lorenz Model

75

(7)

The Billiard Problem

82

(6)

Behavior in the Frequency Domain: Chaos and Noise

88

(6)

The Solar System

94

(35)

Kepler's Laws

94

(7)

The Inverse-Square Law and the Stability of Planetary Orbits

101

(6)

Precession of the Perihelion of Mercury

107

(6)

The Three-Body Problem and the Effect of Jupiter on Earth

113

(5)

Resonances in the Solar System: Kirkwood Gaps and Planetary Rings

118

(5)

Chaotic Tumbling of Hyperion

123

(6)

Potentials and Fields

129

(27)

Electric Potentials and Fields: Laplace's Equation

129

(14)

Potentials and Fields Near Electric Charges

143

(5)

Magnetic Field Produced by a Current

148

(3)

Magnetic Field of a Solenoid: Inside and Out

151

(5)

Waves

156

(25)

Waves: The Ideal Case

156

(9)

Frequency Spectrum of Waves on a String

165

(4)

Motion of a (Somewhat) Realistic String

169

(5)

Waves on a String (Again): Spectral Methods

174

(7)

Random Systems

181

(54)

Why Perform Simulations of Random Processes?

181

(2)

Random Walks

183

(5)

Self-Avoiding Walks

188

(7)

Random Walks and Diffusion

195

(6)

Diffusion, Entropy, and the Arrow of Time

201

(5)

Cluster Growth Models

206

(6)

Fractal Dimensionalities of Curves

212

(6)

Percolation

218

(11)

Diffusion on Fractals

229

(6)

Statistical Mechanics, Phase Transitions, and the Ising Model

235

(35)

The Ising Model and Statistical Mechanics

235

(4)

Mean Field Theory

239

(5)

The Monte Carlo Method

244

(2)

The Ising Model and Second-Order Phase Transitions

246

(13)

First-Order Phase Transitions

259

(5)

Scaling

264

(6)

Molecular Dynamics

270

(33)

Introduction to the Method: Properties of a Dilute Gas

270

(15)

The Melting Transition

285

(9)

Equipartition and the Fermi-Pasta-Ulam Problem

294

(9)

Quantum Mechanics

303

(54)

Time-Independent Schrodinger Equation: Some Preliminaries

303

(4)

One Dimension: Shooting and Matching Methods

307

(16)

A Matrix Approach

323

(3)

A Variational Approach

326

(7)

Time-Dependent Schrodinger Equation: Direct Solutions

333

(12)

Time-Dependent Schrodinger Equation in Two Dimensions

345

(4)

Spectral Methods

349

(8)

Vibrations, Waves, and the Physics of Musical Instruments

357

(32)

Plucking a String: Simulating a Guitar

357

(5)

Striking a String: Pianos and Hammers

362

(5)

Exciting a Vibrating System with Friction: Violins and Bows

367

(5)

Vibrations of a Membrane: Normal Modes and Eigenvalue Problems

372

(10)

Generation of Sound

382

(7)

Interdisciplinary Topics

389

(67)

Protein Folding

389

(16)

Earthquakes and Self-Organized Criticality

405

(13)

Neural Networks and the Brain

418

(18)

Real Neurons and Action Potentials

436

(9)

Cellular Automata

445

(11)

APPENDICES

A Ordinary Differential Equations with Initial Values

456

(13)

A.1 First-Order, Ordinary Differential Equations

456

(4)

A.2 Second-Order, Ordinary Differential Equations

460

(4)

A.3 Centered Difference Methods

464

(3)

A.4 Summary

467

(2)

B Root Finding and Optimization

469

(10)

B.1 Root Finding

469

(3)

B.2 Direct Optimization

472

(1)

B.3 Stochastic Optimization

473

(6)

C The Fourier Transform

479

(14)

C.1 Theoretical Background

479

(2)

C.2 Discrete Fourier Transform

481

(2)

C.3 Fast Fourier Transform (FFT)

483

(3)

C.4 Examples: Sampling Interval and Number of Data Points

486

(2)

C.5 Examples: Aliasing

488

(2)

C.6 Power Spectrum

490

(3)

D Fitting Data to a Function

493

(7)

D.1 Introduction

493

(1)

D.2 Method of Least Squares: Linear Regression for Two Variables

494

(3)

D.3 Method of Least Squares: More General Cases

497

(3)

E Numerical Integration

500

(12)

E.1 Motivation

500

(1)

E.2 Newton-Cotes Methods: Using Discrete Panels to Approximate an Integral

500

(4)

E.3 Gaussian Quadrature: Beyond Classic Methods of Numerical Integration

504

(2)

E.4 Monte Carlo Integration

506

(6)

F Generation of Random Numbers

512

(8)

F.1 Linear Congruential Generators

512

(4)

F.2 Nonuniform Random Numbers

516

(4)

G Statistical Tests of Hypotheses

520

(7)

G.1 Central Limit Theorem and the Χ2 Distribution

521

(2)

G.2 Χ2 Test of a Hypothesis

523

(4)

H Solving Linear Systems

527

(14)

H.1 Solving A x = b, b, b ≠ 0

528

(7)

H.1.1 Gaussian Elimination

528

(2)

H.1.2 Gauss-Jordan elimination

530

(1)

H.1.3 LU decomposition

531

(2)

H.1.4 Relaxational method

533

(2)

H.2 Eigenvalues and Eigenfunctions

535

(6)

H.2.1 Approximate Solution of Eigensystems

537

(4)

Index

541

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Computational Physics

Summary

Author Biography

Table of Contents

Supplemental Materials