Opis: Computational Physics - Cristian Bordeianu, Manuel Paez, Rubin Landau

The use of computation and simulation has become an essential part of the scientific process. Being able to transform a theory into an algorithm requires significant theoretical insight, detailed physical and mathematical understanding, and a working level of competency in programming. This upper-division text provides an unusually broad survey of the topics of modern computational physics from a multidisciplinary, computational science point of view. Its philosophy is rooted in learning by doing (assisted by many model programs), with new scientific materials as well as with the Python programming language. Python has become very popular, particularly for physics education and large scientific projects. It is probably the easiest programming language to learn for beginners, yet is also used for mainstream scientific computing, and has packages for excellent graphics and even symbolic manipulations. The text is designed for an upper-level undergraduate or beginning graduate course and provides the reader with the essential knowledge to understand computational tools and mathematical methods well enough to be successful. As part of the teaching of using computers to solve scientific problems, the reader is encouraged to work through a sample problem stated at the beginning of each chapter or unit, which involves studying the text, writing, debugging and running programs, visualizing the results, and the expressing in words what has been done and what can be concluded. Then there are exercises and problems at the end of each chapter for the reader to work on their own (with model programs given for that purpose).Dedication V Preface XIX 1 Introduction 1 1.1 Computational Physics and Computational Science 1 1.2 This Book's Subjects 3 1.3 This Book's Problems 4 1.4 This Book's Language: The Python Ecosystem 8 1.4.1 Python Packages (Libraries) 9 1.4.2 This Book's Packages 10 1.4.3 The EasyWay: Python Distributions (Package Collections) 12 1.5 Python's Visualization Tools 13 1.5.1 Visual (VPython)'s 2D Plots 14 1.5.2 VPython's Animations 17 1.5.3 Matplotlib's 2D Plots 17 1.5.4 Matplotlib's 3D Surface Plots 22 1.5.5 Matplotlib's Animations 24 1.5.6 Mayavi's Visualizations Beyond Plotting 26 1.6 Plotting Exercises 30 1.7 Python's Algebraic Tools 31 2 Computing Software Basics 33 2.1 Making Computers Obey 33 2.2 ProgrammingWarmup 35 2.2.1 Structured and Reproducible Program Design 36 2.2.2 Shells, Editors, and Execution 37 2.3 Python I/O 39 2.4 Computer Number Representations (Theory) 40 2.4.1 IEEE Floating-Point Numbers 41 2.4.2 Python and the IEEE 754 Standard 47 2.4.3 Over and Underflow Exercises 48 2.4.4 Machine Precision (Model) 49 2.4.5 Experiment: Your Machine's Precision 50 2.5 Problem: Summing Series 51 2.5.1 Numerical Summation (Method) 51 2.5.2 Implementation and Assessment 52 3 Errors and Uncertainties in Computations 53 3.1 Types of Errors (Theory) 53 3.1.1 Model for Disaster: Subtractive Cancelation 55 3.1.2 Subtractive Cancelation Exercises 56 3.1.3 Round-off Errors 57 3.1.4 Round-off Error Accumulation 58 3.2 Error in Bessel Functions (Problem) 58 3.2.1 Numerical Recursion (Method) 59 3.2.2 Implementation and Assessment: Recursion Relations 61 3.3 Experimental Error Investigation 62 3.3.1 Error Assessment 65 4 Monte Carlo: Randomness, Walks, and Decays 69 4.1 Deterministic Randomness 69 4.2 Random Sequences (Theory) 69 4.2.1 Random-Number Generation (Algorithm) 70 4.2.2 Implementation: Random Sequences 72 4.2.3 Assessing Randomness and Uniformity 73 4.3 RandomWalks (Problem) 75 4.3.1 Random-Walk Simulation 76 4.3.2 Implementation: RandomWalk 77 4.4 Extension: Protein Folding and Self-Avoiding RandomWalks 79 4.5 Spontaneous Decay (Problem) 80 4.5.1 Discrete Decay (Model) 81 4.5.2 Continuous Decay (Model) 82 4.5.3 Decay Simulation with Geiger Counter Sound 82 4.6 Decay Implementation and Visualization 84 5 Differentiation and Integration 85 5.1 Differentiation 85 5.2 Forward Difference (Algorithm) 86 5.3 Central Difference (Algorithm) 87 5.4 Extrapolated Difference (Algorithm) 87 5.5 Error Assessment 88 5.6 Second Derivatives (Problem) 90 5.6.1 Second-Derivative Assessment 90 5.7 Integration 91 5.8 Quadrature as Box Counting (Math) 91 5.9 Algorithm: Trapezoid Rule 93 5.10 Algorithm: Simpson's Rule 94 5.11 Integration Error (Assessment) 96 5.12 Algorithm: Gaussian Quadrature 97 5.12.1 Mapping Integration Points 98 5.12.2 Gaussian Points Derivation 99 5.12.3 Integration Error Assessment 100 5.13 Higher Order Rules (Algorithm) 103 5.14 Monte Carlo Integration by Stone Throwing (Problem) 104 5.14.1 Stone Throwing Implementation 104 5.15 Mean Value Integration (Theory and Math) 105 5.16 Integration Exercises 106 5.17 Multidimensional Monte Carlo Integration (Problem) 108 5.17.1 Multi Dimension Integration Error Assessment 109 5.17.2 Implementation: 10D Monte Carlo Integration 110 5.18 Integrating Rapidly Varying Functions (Problem) 110 5.19 Variance Reduction (Method) 110 5.20 Importance Sampling (Method) 111 5.21 von Neumann Rejection (Method) 111 5.21.1 Simple Random Gaussian Distribution 113 5.22 Nonuniform Assessment !N 113 5.22.1 Implementation !N 114 6 Matrix Computing 117 6.1 Problem 3: N-D Newton-Raphson; Two Masses on a String 117 6.1.1 Theory: Statics 118 6.1.2 Algorithm: Multidimensional Searching 119 6.2 Why Matrix Computing? 122 6.3 Classes ofMatrix Problems (Math) 122 6.3.1 Practical Matrix Computing 124 6.4 Python Lists as Arrays 126 6.5 Numerical Python (NumPy) Arrays 127 6.5.1 NumPy's linalg Package 132 6.6 Exercise: TestingMatrix Programs 134 6.6.1 Matrix Solution of the String Problem 137 6.6.2 Explorations 139 7 Trial-and-Error Searching and Data Fitting 141 7.1 Problem 1: A Search for Quantum States in a Box 141 7.2 Algorithm: Trial-and-Error Roots via Bisection 142 7.2.1 Implementation: Bisection Algorithm 144 7.3 Improved Algorithm: Newton-Raphson Searching 145 7.3.1 Newton-Raphson with Backtracking 147 7.3.2 Implementation: Newton-Raphson Algorithm 148 7.4 Problem 2: Temperature Dependence ofMagnetization 148 7.4.1 Searching Exercise 150 7.5 Problem 3: Fitting An Experimental Spectrum 150 7.5.1 Lagrange Implementation, Assessment 152 7.5.2 Cubic Spline Interpolation (Method) 153 7.6 Problem 4: Fitting Exponential Decay 156 7.7 Least-Squares Fitting (Theory) 158 7.7.1 Least-Squares Fitting: Theory and Implementation 160 7.8 Exercises: Fitting Exponential Decay,Heat Flow andHubble's Law 162 7.8.1 Linear Quadratic Fit 164 7.8.2 Problem 5: Nonlinear Fit to a Breit-Wigner 167 8 Solving Differential Equations: Nonlinear Oscillations 171 8.1 Free Nonlinear Oscillations 171 8.2 Nonlinear Oscillators (Models) 171 8.3 Types of Differential Equations (Math) 173 8.4 Dynamic Form for ODEs (Theory) 175 8.5 ODE Algorithms 177 8.5.1 Euler's Rule 177 8.6 Runge-Kutta Rule 178 8.7 Adams-Bashforth-Moulton Predictor-Corrector Rule 183 8.7.1 Assessment: rk2 vs. rk4 vs. rk45 185 8.8 Solution for Nonlinear Oscillations (Assessment) 187 8.8.1 Precision Assessment: Energy Conservation 188 8.9 Extensions: Nonlinear Resonances, Beats, Friction 189 8.9.1 Friction (Model) 189 8.9.2 Resonances and Beats: Model, Implementation 190 8.10 Extension: Time-Dependent Forces 190 9 ODE Applications: Eigenvalues, Scattering, and Projectiles 193 9.1 Problem: Quantum Eigenvalues in Arbitrary Potential 193 9.1.1 Model: Nucleon in a Box 194 9.2 Algorithms: Eigenvalues via ODE Solver + Search 195 9.2.1 Numerov Algorithm for Schrodinger ODE !N 197 9.2.2 Implementation: Eigenvalues viaODESolver + BisectionAlgorithm 200 9.3 Explorations 203 9.4 Problem: Classical Chaotic Scattering 203 9.4.1 Model and Theory 204 9.4.2 Implementation 206 9.4.3 Assessment 207 9.5 Problem: Balls Falling Out of the Sky 208 9.6 Theory: Projectile Motion with Drag 208 9.6.1 Simultaneous Second-Order ODEs 209 9.6.2 Assessment 210 9.7 Exercises: 2- and 3-Body Planet Orbits and ChaoticWeather 211 10 High-Performance Hardware and Parallel Computers 215 10.1 High-Performance Computers 215 10.2 Memory Hierarchy 216 10.3 The Central Processing Unit 219 10.4 CPU Design: Reduced Instruction Set Processors 220 10.5 CPU Design:Multiple-Core Processors 221 10.6 CPU Design: Vector Processors 222 10.7 Introduction to Parallel Computing 223 10.8 Parallel Semantics (Theory) 224 10.9 Distributed Memory Programming 226 10.10 Parallel Performance 227 10.10.1 Communication Overhead 229 10.11 Parallelization Strategies 230 10.12 Practical Aspects of MIMD Message Passing 231 10.12.1 High-Level View ofMessage Passing 233 10.12.2 Message Passing Example and Exercise 234 10.13 Scalability 236 10.13.1 Scalability Exercises 238 10.14 Data Parallelism and Domain Decomposition 239 10.14.1 Domain Decomposition Exercises 242 10.15 Example: The IBM Blue Gene Supercomputers 243 10.16 Exascale Computing via Multinode-Multicore GPUs 245 11 Applied HPC: Optimization, Tuning, and GPU Programming 247 11.1 General Program Optimization 247 11.1.1 Programming for Virtual Memory (Method) 248 11.1.2 Optimization Exercises 249 11.2 Optimized Matrix Programming with NumPy 251 11.2.1 NumPy Optimization Exercises 254 11.3 Empirical Performance of Hardware 254 11.3.1 Racing Python vs. Fortran/C 255 11.4 Programming for the Data Cache (Method) 262 11.4.1 Exercise 1: Cache Misses 264 11.4.2 Exercise 2: Cache Flow 264 11.4.3 Exercise 3: Large-Matrix Multiplication 265 11.5 Graphical Processing Units for High Performance Computing 266 11.5.1 The GPU Card 267 11.6 Practical Tips forMulticore and GPU Programming !N 267 11.6.1 CUDA Memory Usage 270 11.6.2 CUDA Programming !N 271 12 Fourier Analysis: Signals and Filters 275 12.1 Fourier Analysis of Nonlinear Oscillations 275 12.2 Fourier Series (Math) 276 12.2.1 Examples: Sawtooth and Half-Wave Functions 278 12.3 Exercise: Summation of Fourier Series 279 12.4 Fourier Transforms (Theory) 279 12.5 The Discrete Fourier Transform 281 12.5.1 Aliasing (Assessment) 285 12.5.2 Fourier Series DFT (Example) 287 12.5.3 Assessments 288 12.5.4 Nonperiodic Function DFT (Exploration) 290 12.6 Filtering Noisy Signals 290 12.7 Noise Reduction via Autocorrelation (Theory) 290 12.7.1 Autocorrelation Function Exercises 293 12.8 Filtering with Transforms (Theory) 294 12.8.1 Digital Filters:Windowed Sinc Filters (Exploration) !N 296 12.9 The Fast Fourier Transform Algorithm !N 299 12.9.1 Bit Reversal 301 12.10 FFT Implementation 303 12.11 FFT Assessment 304 13 Wavelet and Principal Components Analyses: Nonstationary Signals and Data Compression 307 13.1 Problem: Spectral Analysis of Nonstationary Signals 307 13.2 Wavelet Basics 307 13.3 Wave Packets and Uncertainty Principle (Theory) 309 13.3.1 Wave Packet Assessment 311 13.4 Short-Time Fourier Transforms (Math) 311 13.5 TheWavelet Transform 313 13.5.1 GeneratingWavelet Basis Functions 313 13.5.2 ContinuousWavelet Transform Implementation 316 13.6 DiscreteWavelet Transforms,Multiresolution Analysis !N 317 13.6.1 Pyramid Scheme Implementation !N 323 13.6.2 DaubechiesWavelets via Filtering 327 13.6.3 DWT Implementation and Exercise 330 13.7 Principal Components Analysis 332 13.7.1 Demonstration of Principal Component Analysis 334 13.7.2 PCA Exercises 337 14 Nonlinear Population Dynamics 339 14.1 Bug Population Dynamics 339 14.2 The Logistic Map (Model) 339 14.3 Properties of NonlinearMaps (Theory and Exercise) 341 14.3.1 Fixed Points 342 14.3.2 Period Doubling, Attractors 343 14.4 Mapping Implementation 344 14.5 Bifurcation Diagram (Assessment) 345 14.5.1 Bifurcation Diagram Implementation 346 14.5.2 Visualization Algorithm: Binning 347 14.5.3 Feigenbaum Constants (Exploration) 348 14.6 Logistic Map Random Numbers (Exploration) !N 348 14.7 Other Maps (Exploration) 348 14.8 Signals of Chaos: Lyapunov Coefficient and Shannon Entropy !N 349 14.9 Coupled Predator-PreyModels 353 14.10 Lotka-Volterra Model 354 14.10.1 Lotka-Volterra Assessment 356 14.11 Predator-Prey Chaos 356 14.11.1 Exercises 359 14.11.2 LVM with Prey Limit 359 14.11.3 LVM with Predation Efficiency 360 14.11.4 LVM Implementation and Assessment 361 14.11.5 Two Predators, One Prey (Exploration) 362 15 Continuous Nonlinear Dynamics 363 15.1 Chaotic Pendulum 363 15.1.1 Free Pendulum Oscillations 364 15.1.2 Solution as Elliptic Integrals 365 15.1.3 Implementation and Test: Free Pendulum 366 15.2 Visualization: Phase-Space Orbits 367 15.2.1 Chaos in Phase Space 368 15.2.2 Assessment in Phase Space 372 15.3 Exploration: Bifurcations of Chaotic Pendulums 374 15.4 Alternate Problem: The Double Pendulum 375 15.5 Assessment: Fourier/Wavelet Analysis of Chaos 377 15.6 Exploration: Alternate Phase-Space Plots 378 15.7 Further Explorations 379 16 Fractals and Statistical Growth Models 383 16.1 Fractional Dimension (Math) 383 16.2 The Sierpi..1/2ski Gasket (Problem 1) 384 16.2.1 Sierpi..1/2ski Implementation 384 16.2.2 Assessing Fractal Dimension 385 16.3 Growing Plants (Problem 2) 386 16.3.1 Self-Affine Connection (Theory) 386 16.3.2 Barnsley's Fern Implementation 387 16.3.3 Self-Affinity in Trees Implementation 389 16.4 Ballistic Deposition (Problem 3) 390 16.4.1 Random Deposition Algorithm 390 16.5 Length of British Coastline (Problem 4) 391 16.5.1 Coastlines as Fractals (Model) 392 16.5.2 Box Counting Algorithm 392 16.5.3 Coastline Implementation and Exercise 393 16.6 Correlated Growth, Forests, Films (Problem 5) 395 16.6.1 Correlated Ballistic Deposition Algorithm 395 16.7 Globular Cluster (Problem 6) 396 16.7.1 Diffusion-Limited Aggregation Algorithm 396 16.7.2 Fractal Analysis of DLA or a Pollock 399 16.8 Fractals in Bifurcation Plot (Problem 7) 400 16.9 Fractals from Cellular Automata 400 16.10 Perlin Noise Adds Realism !N 402 16.10.1 Ray Tracing Algorithms 404 16.11 Exercises 407 17 Thermodynamic Simulations and Feynman Path Integrals 409 17.1 Magnets via Metropolis Algorithm 409 17.2 An IsingChain (Model) 410 17.3 Statistical Mechanics (Theory) 412 17.3.1 Analytic Solution 413 17.4 Metropolis Algorithm 413 17.4.1 Metropolis Algorithm Implementation 416 17.4.2 Equilibration, Thermodynamic Properties (Assessment) 417 17.4.3 Beyond Nearest Neighbors, 1D (Exploration) 419 17.5 Magnets viaWang-Landau Sampling !N 420 17.6 Wang-Landau Algorithm 423 17.6.1 WLS IsingModel Implementation 425 17.6.2 WLS IsingModel Assessment 428 17.7 Feynman Path Integral Quantum Mechanics !N 429 17.8 Feynman's Space-Time Propagation (Theory) 429 17.8.1 Bound-StateWave Function (Theory) 431 17.8.2 Lattice Path Integration (Algorithm) 432 17.8.3 Lattice Implementation 437 17.8.4 Assessment and Exploration 440 17.9 Exploration: Quantum Bouncer's Paths !N 440 18 Molecular Dynamics Simulations 445 18.1 Molecular Dynamics (Theory) 445 18.1.1 Connection to Thermodynamic Variables 449 18.1.2 Setting Initial Velocities 449 18.1.3 Periodic Boundary Conditions and Potential Cutoff 450 18.2 Verlet and Velocity-Verlet Algorithms 451 18.3 1D Implementation and Exercise 453 18.4 Analysis 456 19 PDE Reviewand Electrostatics via Finite Differences and Electrostatics via Finite Differences 461 19.1 PDE Generalities 461 19.2 Electrostatic Potentials 463 19.2.1 Laplace's Elliptic PDE (Theory) 463 19.3 Fourier Series Solution of a PDE 464 19.3.1 Polynomial Expansion as an Algorithm 466 19.4 Finite-Difference Algorithm 467 19.4.1 Relaxation and Over-relaxation 469 19.4.2 Lattice PDE Implementation 470 19.5 Assessment via Surface Plot 471 19.6 Alternate Capacitor Problems 471 19.7 Implementation and Assessment 474 19.8 Electric Field Visualization (Exploration) 475 19.9 Review Exercise 476 20 Heat Flow via Time Stepping 477 20.1 Heat Flow via Time-Stepping (Leapfrog) 477 20.2 The Parabolic Heat Equation (Theory) 478 20.2.1 Solution: Analytic Expansion 478 20.2.2 Solution: Time Stepping 479 20.2.3 von Neumann Stability Assessment 481 20.2.4 Heat Equation Implementation 483 20.3 Assessment and Visualization 483 20.4 Improved Heat Flow: Crank-Nicolson Method 484 20.4.1 Solution of Tridiagonal Matrix Equations !N 487 20.4.2 Crank-Nicolson Implementation, Assessment 490 21 Wave Equations I: Strings and Membranes 491 21.1 A Vibrating String 491 21.2 The HyperbolicWave Equation (Theory) 491 21.2.1 Solution via Normal-Mode Expansion 493 21.2.2 Algorithm: Time Stepping 494 21.2.3 Wave Equation Implementation 496 21.2.4 Assessment, Exploration 497 21.3 Strings with Friction (Extension) 499 21.4 Strings with Variable Tension and Density 500 21.4.1 Waves on Catenary 501 21.4.2 Derivation of Catenary Shape 501 21.4.3 Catenary and FrictionalWave Exercises 503 21.5 Vibrating Membrane (2DWaves) 504 21.6 Analytical Solution 505 21.7 Numerical Solution for 2DWaves 508 22 Wave Equations II: QuantumPackets and Electromagnetic 511 22.1 Quantum Wave Packets 511 22.2 Time-Dependent Schrodinger Equation (Theory) 511 22.2.1 Finite-Difference Algorithm 513 22.2.2 Wave Packet Implementation, Animation 514 22.2.3 Wave Packets in OtherWells (Exploration) 516 22.3 Algorithm for the 2D Schrodinger Equation 517 22.3.1 Exploration: Bound and Diffracted 2D Packet 518 22.4 Wave Packet-Wave Packet Scattering 518 22.4.1 Algorithm 520 22.4.2 Implementation 520 22.4.3 Results and Visualization 522 22.5 E&MWaves via Finite-Difference Time Domain 525 22.6 Maxwell's Equations 525 22.7 FDTD Algorithm 526 22.7.1 Implementation 530 22.7.2 Assessment 530 22.7.3 Extension: Circularly PolarizedWaves 531 22.8 Application:Wave Plates 533 22.9 Algorithm 534 22.10 FDTD Exercise and Assessment 535 23 Electrostatics via Finite Elements 537 23.1 Finite-Element Method !N 537 23.2 Electric Field from Charge Density (Problem) 538 23.3 Analytic Solution 538 23.4 Finite-Element (Not Difference) Methods, 1D 539 23.4.1 Weak Form of PDE 539 23.4.2 Galerkin Spectral Decomposition 540 23.5 1D FEMImplementation and Exercises 544 23.5.1 1D Exploration 547 23.6 Extension to 2D Finite Elements 547 23.6.1 Weak Form of PDE 548 23.6.2 Galerkin's Spectral Decomposition 548 23.6.3 Triangular Elements 549 23.6.4 Solution as Linear Equations 551 23.6.5 Imposing Boundary Conditions 552 23.6.6 FEM2D Implementation and Exercise 554 23.6.7 FEM2D Exercises 554 24 Shocks Waves and Solitons 555 24.1 Shocks and Solitons in ShallowWater 555 24.2 Theory: Continuity and Advection Equations 556 24.2.1 Advection Implementation 558 24.3 Theory: ShockWaves via Burgers' Equation 559 24.3.1 Lax-Wendroff Algorithm for Burgers' Equation 560 24.3.2 Implementation and Assessment of Burgers' Shock Equation 561 24.4 Including Dispersion 562 24.5 Shallow-Water Solitons: The KdeV Equation 563 24.5.1 Analytic Soliton Solution 563 24.5.2 Algorithm for KdeV Solitons 564 24.5.3 Implementation: KdeV Solitons 565 24.5.4 Exploration: Solitons in Phase Space, Crossing 567 24.6 Solitons on Pendulum Chain 567 24.6.1 Including Dispersion 568 24.6.2 Continuum Limit, the Sine-Gordon Equation 570 24.6.3 Analytic SGE Solution 571 24.6.4 Numeric Solution: 2D SGE Solitons 571 24.6.5 2D Soliton Implementation 573 24.6.6 SGE Soliton Visualization 574 25 Fluid Dynamics 575 25.1 River Hydrodynamics 575 25.2 Navier-Stokes Equation (Theory) 576 25.2.1 Boundary Conditions for Parallel Plates 578 25.2.2 Finite-Difference Algorithm and Overrelaxation 580 25.2.3 Successive Overrelaxation Implementation 581 25.3 2D Flow over a Beam 581 25.4 Theory: Vorticity Form of Navier-Stokes Equation 582 25.4.1 Finite Differences and the SOR Algorithm 584 25.4.2 Boundary Conditions for a Beam 585 25.4.3 SOR on a Grid 587 25.4.4 Flow Assessment 589 25.4.5 Exploration 590 26 Integral Equations of QuantumMechanics 591 26.1 Bound States of Nonlocal Potentials 591 26.2 Momentum-Space Schrodinger Equation (Theory) 592 26.2.1 Integral toMatrix Equations 593 26.2.2 Delta-Shell Potential (Model) 595 26.2.3 Binding Energies Solution 595 26.2.4 Wave Function (Exploration) 597 26.3 Scattering States of Nonlocal Potentials !N 597 26.4 Lippmann-Schwinger Equation (Theory) 598 26.4.1 Singular Integrals (Math) 599 26.4.2 Numerical Principal Values 600 26.4.3 Reducing Integral Equations to Matrix Equations (Method) 600 26.4.4 Solution via Inversion, Elimination 602 26.4.5 Scattering Implementation 603 26.4.6 ScatteringWave Function (Exploration) 604 Appendix A Codes, Applets, and Animations 607 Bibliography 609 Index 615

Szczegóły: Computational Physics - Cristian Bordeianu, Manuel Paez, Rubin Landau

Tytuł: Computational Physics
Autor: Cristian Bordeianu, Manuel Paez, Rubin Landau
Producent: VCH
ISBN: 9783527413157
Rok produkcji: 2015
Ilość stron: 644
Oprawa: Miękka
Waga: 1.37 kg

Recenzje: Computational Physics - Cristian Bordeianu, Manuel Paez, Rubin Landau