Computational physics

Contents......Page 6
Preface......Page 10
About the Authors......Page 13
1.1    Radioactive Decay......Page 16
1.2    A Numerical Approach......Page 17
1.3    Design and Construction of a Working Program: Codes and Pseudocodes......Page 18
1.4    Testing Your Program......Page 26
1.5    Numerical Considerations......Page 27
1.6    Programming Guidelines and Philosophy......Page 29
2.1    Bicycle Racing: The Effect of Air Resistance......Page 33
2.2    Projectile Motion. The Trajectory of a Cannon Shell......Page 40
2.3    Baseball: Motion of a Batted Ball......Page 46
2.4    Throwing a Baseball: The Effects of Spin......Page 51
2.5    Golf......Page 59
3.1    Simple Harmonic Motion......Page 63
3.2    Making the Pendulum More Interesting: Adding Dissipation, Non-linearity, and a Driving Force......Page 69
3.3    Chaos in the Driven Nonlinear Pendulum......Page 73
3.4    Routes to Chaos: Period Doubling......Page 81
3.5    The Logistic Map: Why the Period Doubles......Page 85
3.6    The Lorenz Model......Page 90
3.7    The Billiard Problem......Page 97
3.8    Behavior in the Frequency Domain: Chaos and Noise......Page 103
4.1    Kepler's Laws......Page 107
4.2    The Inverse-Square Law and the Stability of Planetary Orbits......Page 114
4.3    Precession of the Perihelion of Mercury......Page 120
4.4    The Three-Body Problem and the Effect of Jupiter on Earth......Page 126
4.5    Resonances in the Solar System: Kirkwood Gaps and Planetary Rings......Page 131
4.6    Chaotic Tumbling of Hyperion......Page 136
5.1    Electric Potentials and Fields: Laplace's Equation......Page 140
5.2    Potentials and Fields Near Electric Charges......Page 154
5.3    Magnetic Field Produced by a Current......Page 159
5.4    Magnetic Field of a Solenoid: Inside and Out......Page 162
6.1    Waves: The Ideal Case......Page 167
6.2    Frequency Spectrum of Waves on a String......Page 176
6.3    Motion of a (Somewhat) Realistic String......Page 180
6.4    Waves on a String (Again): Spectral Methods......Page 185
7.1    Why Perform Simulations of Random Processes?......Page 192
7.2    Random Walks......Page 194
7.3    Self-Avoiding Walks......Page 199
7.4    Random Walks and Diffusion......Page 206
7.5    Diffusion, Entropy, and the Arrow of Time......Page 212
7.6    Cluster Growth Models......Page 217
7.7    Fractal Dimensionalities of Curves......Page 223
7.8    Percolation......Page 229
7.9    Diffusion on Fractals......Page 240
8.1    The Ising Model and Statistical Mechanics......Page 246
8.2    Mean Field Theory......Page 250
8.3    The Monte Carlo Method......Page 255
8.4    The Ising Model and Second-Order Phase Transitions......Page 257
8.5    First-Order Phase Transitions......Page 270
8.6    Scaling......Page 275
9.1    Introduction to the Method: Properties of a Dilute Gas......Page 281
9.2    The Melting Transition......Page 296
9.3    Equipartition and the Fermi-Pasta-Ulam Problem......Page 305
10  Quantum Mechanics......Page 314
10.1  Time-Independent Schrodinger Equation: Some Preliminaries......Page 
10.2  One Dimension: Shooting and Matching Methods......Page 318
10.3  A Matrix Approach......Page 334
10.4  A Variational Approach......Page 337
10.5  Time-Dependent Schrodinger Equation: Direct Solutions......Page 344
10.6  Time-Dependent Schrodinger Equation in Two Dimensions......Page 356
10.7  Spectral Methods......Page 360
11.1  Plucking a String: Simulating a Guitar......Page 368
11.2  Striking a String: Pianos and Hammers......Page 373
11.3  Exciting a Vibrating System with Friction: Violins and Bows......Page 378
11.4  Vibrations of a Membrane: Normal Modes and Eigenvalue Problems......Page 383
11.5  Generation of Sound......Page 393
12.1  Protein Folding......Page 400
12.2  Earthquakes and Self-Organized Criticality......Page 416
12.3  Neural Networks and the Brain......Page 429
12.4  Real Neurons and Action Potentials......Page 447
12.5  Cellular Automata......Page 456
A.4  Summary......Page 467
A.2  Second-Order, Ordinary Differential Equations......Page 471
A.3  Centered Difference Methods......Page 475
B.1   Root Finding......Page 480
B.2   Direct Optimization......Page 483
B.3   Stochastic Optimization......Page 484
C.1   Theoretical Background......Page 490
C.2   Discrete Fourier Transform......Page 492
C.3   Fast Fourier Transform (FFT)......Page 494
C.4  Examples: Sampling Interval and Number of Data Points......Page 497
C.5   Examples: Aliasing......Page 499
C.6  Power Spectrum......Page 501
D.1  Introduction......Page 504
D.2  Method of Least Squares: Linear Regression for Two Variables......Page 505
D.3  Method of Least Squares: More General Cases......Page 508
E.2   Newton-Cotes Methods: Using Discrete Panels to Approximate an Integral......Page 511
E.3   Gaussian Quadrature: Beyond Classic Methods of Numerical Integration......Page 515
E.4   Monte Carlo Integration......Page 517
F.1   Linear Congruential Generators......Page 523
F.2   Nonuniform Random Numbers......Page 527
G Statistical Tests of Hypotheses......Page 531
G.1   Central Limit Theorem and the y2 Distribution......Page 532
G.2   y2 Test of a Hypothesis......Page 534
H Solving Linear Systems......Page 538
H.1.1   Gaussian Elimination......Page 539
H.1.2   Gauss-Jordan elimination......Page 541
H.1.3   LU decomposition......Page 542
H.1.4   Relaxational method......Page 544
H.2   Eigenvalues and Eigenfunctions......Page 546
H.2.1   Approximate Solution of Eigensystems......Page 548
Index......Page 552