Computational Physics - Cristian C. Bordeianu

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Résumé :

Preface xvii

Acknowledgments xix

Part I Basics 1

1 Introduction 3

1.1 Computational Physics and Science 3

1.2 This Book's Subjects 4

1.3 Video Lecture Supplements 4

1.4 This Book's Codes and Problems 5

1.5 Our Language: The Python Ecosystem 6

1.6 The Easy Way: Python Distributions 6

2 Software Basics 9

2.1 Making Computers Obey 9

2.2 Computer Number Representations 11

2.3 Python Mini Tutorial 18

2.4 Programming Warmup 25

2.5 Python's Visualization Tools 30

2.6 Plotting Exercises 36

2.7 Code Listings 38

3 Errors and Uncertainties 44

3.1 Types of Errors 44

3.2 Experimental Error Investigation 49

3.3 Errors with Power Series 52

3.4 Errors in Bessel Functions 55

3.5 Code Listing 58

4 Monte Carlo Simulations 59

4.1 Random Numbers 59

4.2 Simulating a Random Walk 63

4.3 Spontaneous Decay 68

4.4 Testing and Generating Random Distributions 71

4.5 Code Listings 73

5 Differentiation and Integration 78

5.1 Differentiation Algorithms 78

5.2 Extrapolated Difference 80

5.3 Integration Algorithms 83

5.4 Gaussian Quadrature 89

5.5 Monte Carlo Integrations 91

5.6 Mean Value and N-D Integration 94

5.7 mc Variance Reduction 96

5.8 Importance Sampling and von Neumann Rejection 96

5.9 Code Listings 97

6 Trial-and-Error Searching and Data Fitting 100

6.1 Quantum Bound States I 100

6.2 Bisection Search 101

6.3 Newton-Raphson Search 102

6.4 Magnetization Search 105

6.5 Data Fitting 107

6.6 Fitting Exponential Decay 112

6.7 Least-Squares Fitting 113

6.8 Nonlinear Fit to a Resonance 118

6.9 Code Listings 120

7 Matrix Computing and N-D Searching 123

7.1 Masses on a String and N-D Searching 123

7.2 Matrix Generalities 126

7.3 Matrices in Python 129

7.4 Exercise: Tests Before Use 136

7.5 Solution to String Problem 139

7.6 Spin States and Hyperfine Structure 139

7.7 Speeding Up Matrix Computing ? 141

7.8 Code Listing 144

8 Differential Equations and Nonlinear Oscillations 147

8.1 Nonlinear Oscillators 147

8.2 ODE Review 149

8.3 Dynamic Form of ODEs 150

8.4 ODE Algorithms 152

8.5 Solution for Nonlinear Oscillations 157

8.6 Extensions: Nonlinear Resonances, Beats, Friction 159

8.7 Code Listings 161

Part II Data Science 165

9 Fourier Analyses 167

9.1 Fourier Series 167

9.2 Fourier Transforms 170

9.3 Discrete Fourier Transforms 172

9.4 Noise Filtering 178

9.5 Fast Fourier Transform ? 185

9.6 FFT Implementation 189

9.7 FFT Assessment 190

9.8 Code Listings 190

10 Wavelet and Principal Components Analysis 193

10.1 Part I: Wavelet Analysis 193

10.2 Wave Packets and Uncertainty Principle 195

10.3 Short-Time Fourier Transforms 197

10.4 Wavelet Transforms 198

10.5 Discrete Wavelet Transforms ? 203

10.6 Part II: Principal Components Analysis 213

10.7 Code Listings 220

11 Neural Networks and Machine Learning 224

11.1 Part I: Biological and Artificial Neural Networks 225

11.2 A Simple Neural Network 226

11.3 A Graphical Deep Net 232

11.4 Part II: Machine Learning Software 234

11.5 TensorFlow and SkLearn Examples 235

11.6 ml Clustering 240

11.7 Keras: Python's Deep Learning API 244

11.8 Image Processing with OpenCV 244

11.9 Explore ML Dat...

Biographie:

The classic in the field for more than 25 years, now with increased emphasis on data science and new chapters on quantum computing, machine learning (AI), and general relativity

Computational physics combines physics, applied mathematics, and computer science in a cutting-edge multidisciplinary approach to solving realistic physical problems. It has become integral to modern physics research because of its capacity to bridge the gap between mathematical theory and real-world system behavior.

Computational Physics provides the reader with the essential knowledge to understand computational tools and mathematical methods well enough to be successful. Its philosophy is rooted in learning by doing, assisted by many sample programs in the popular Python programming language. The first third of the book lays the fundamentals of scientific computing, including programming basics, stable algorithms for differentiation and integration, and matrix computing. The latter two-thirds of the textbook cover more advanced topics such linear and nonlinear differential equations, chaos and fractals, Fourier analysis, nonlinear dynamics, and finite difference and finite elements methods. A particular focus in on the applications of these methods for solving realistic physical problems.

Readers of the fourth edition of Computational Physics will also find:

• An exceptionally broad range of topics, from simple matrix manipulations to intricate computations in nonlinear dynamics
• A whole suite of supplementary material: Python programs, Jupyter notebooks and videos

Computational Physics is ideal for students in physics, engineering, materials science, and any subjects drawing on applied physics....

Sommaire:
Eigenvalues, Scattering, Trajectories 291

13.1 Quantum Eigenvalues for Arbitrary Potentials 291

13.2 Algorithm: ODE Solver + Search 293

13.3 Classical Chaotic Scattering 296

13.4 Projectile Motion with Drag 299

13.5 2- and 3-Body Planetary Orbits 301

13.6 Code Listings 303

14 Fractals and Statistical Growth Models 307

14.1 The Sierpi?ski Gasket 308

14.2 Growing Plants 310

14.3 Ballistic Deposition 312

14.4 Length of British Coastline 313

14.5 Correlated Growth 317

14.6 Diffusion-Limited Aggregation 318

14.7 Fractals in Bifurcations 320

14.8 Cellular Automata Fractals 320

14.9 Perlin Noise Adds Realism ? 321

14.10 Code Listings 324

15 Nonlinear Population Dynamics 329

15.1 The Logistic Map, A Bug Population Model 329

15.2 Chaos 333

15.3 Bifurcation Diagrams 333

15.4 Measures of Chaos 336

15.5 Coupled Predator-Prey Models ? 338

15.6 Code Listings 344

16 Nonlinear Dynamics of Continuous Systems 348

16.1 The Chaotic Pendulum 348

16.2 Phase Space 351

16.3 Chaotic Explorations 354

16.4 Other Chaotic Systems 358

16.5 Code Listings 364

17 Thermodynamics Simulations and Feynman Path Integrals 365

17.1 An Ising Magnetic Chain 365

17.2 Metropolis Algorithm 368

17.3 Fast Equilibration via Wang-Landau Sampling ? 372

17.4 Path Integral Quantum Mechanics ? 374

17.5 Lattice Path Integration 377

17.6 Implementation 381

17.7 Code Listings 385

18 Molecular Dynamics Simulations 391

18.1 MD Versus Thermodynamics 394

18.2 Initial, Boundary, and Large r Conditions 394

18.3 Verlet Algorithms 396

18.4 MD for 16 Particles 400

18.5 Code Listing 402

19 General Relativity 408

19.1 Einstein's Field Equations 408

19.2 Gravitational Deflection of Light 412

19.3 Planetary Orbits in GR Gravity 414

19.4 Visualizing Wormholes 418

19.5 Problems 420

19.6 Code Listings 420

20 Integral Equations 425

20.1 Nonlocal Potential Binding 425

20.2 Momentum-Space Schr?dinger Equation 425

20.3 Scattering in Momentum Space ? 429

20.4 Code Listings 434

Part IV PDE Applications 437

21 PDE Review, Electrostatics and Relaxation 439

21.1 Review 439

21.2 Laplace's Equation 441

21.3 Finite-Difference Algorithm 444

21.4 Alternate Capacitor Problems 447

21.5 Electric Field Visualization 449

21.6 Code Listings 450

22 Heat Flow and Leapfrogging 452

22.1 The Parabolic Heat Equation 452

22.2 Time Stepping (Leapfrog) Algorithm 454

22.3 Newton's Radiative Cooling 457

22.4 The Crank-Nicolson Algorithm 458

22.5 Code Listings 462

23 String and Membrane Waves 464

23.1 A Vibrating String's Hyperbolic Wave Equation 464

23.2 Time-Stepping Algorithm 466

23.3 von Neumann Stability Analysis 468

23.4 Beyond The Simple Wave Equation 469

23.5 Vibrating Membrane (2D Waves) 474

23.6 Analytical Solution 475

23.7 Numerical Solution 476

23.8 Code Listings 478

24 Quantum Wave Packets and EM Waves 480

24.1 Time-Dependent Schr?dinger Equation 480

24.2 Split-Time Algorithm 482

24.3 Special Schr?dinger Algorithm 484

24.4 Quantum Chaos 485

24.5 E&M Waves: Finite Difference Time Domain 488

24.6 Maxwell's Equations 488

24.7 Split-Time FDTD 489

24.8 More E&M Problems 492

24.9 Code Listings 496

25 Shock and Soliton...