An Introduction to Computer Simulation Methods
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1 An Introduction to Computer Simulation Methods Applications to Physical Systems Second Edition Harvey Gould Department of Physics Clark University Jan Tobochnik Department of Physics Kalamazoo College Addison-Wesley Publishing Company Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan Milan Paris
2 C O N T E N T S Preface v 1 Introduction Importance of Computers in Physics The Nature of Computer Simulation Importance of Graphics Programming Languages Learning to Program How to Use This Book 7 1A Laboratory Report 8 2 The Coffee Cooling Problem Background The Euler Algorithm A Simple Example Some True BASIC Programs A Computer Program for the Euler Method The Coffee Cooling Problem Accuracy and Stability Simple Plots Visualization NuclearDecay Overview 34 2A Integer and Real Variables 35 3 The Motion of Falling Objects Background The Force on a Falling Object The Euler Method for Newton's Laws of Motion A Program for One-Dimensional Motion Two-Dimensional Trajectories 48 ix
3 X Contents 3.6 Levels of Simulation Further Applications 51 3A Subroutine for Drawing Axes 51 3B Data Files 52 3C Strong Typing and Debugging 53 3D The Euler-Richardson Method 56 The Two-Body Problem Introduction The Equationsof Motion Circular and Elliptical Orbits Astronomical Units Array Variables and Aspect Ratio Log-log and Semilog Plots Simulation of the Orbit Perturbations Velocity Space A Mini-Solar System Two-Body Scattering Projects 89 Simple Linear and Nonlinear Systems Simple Harmonie Motion Numerical Simulation of the Harmonie Oscillator The Simple Pendulum Output and Animation Dissipative Systems Response to External Forces Electrical Circuit Oscillations Projects 118 5A Numerical Integration of Newton's Equation of Motion 120 The Chaotic Motion of Dynamical Systems Introduction A Simple One-Dimensional Map Period-Doubling Universal Properties and Self-Similarity Measuring Chaos Controlling Chaos Higher-Dimensional Models Forced Damped Pendulum 156
4 Contents xi 6.9 Hamiltonian Chaos Perspective Projects 170 6A Stability of the Fixed Points of the Logistic Map Random Processes Order to Disorder The Poisson Distribution and Nuclear Decay Introduction to Random Walks Problems in Probability Method of Least Squares A Simple Variational Monte Carlo Method 206 7A Random Walks and the Diffusion Equation The Dynamics of Many Particie Systems Introduction The Intermolecular Potential The Numerical Algorithm Boundary Conditions Units A Molecular Dynamics Program Thermodynamic Quantities Radial Distribution Function Hard disks Dynamical Properties Extensions Projects Normal Modes and Waves Coupled Oscillators and Normal Modes Fourier Transforms Wave Motion Interference and Diffraction 292 9A Fast Fourier Transform Electrodynamics Static Charges Numerical Solutions of Laplace's Equation Random Walk Solution of Laplace's Equation Fields Due to Moving Charges 321
5 xii Contents 10.5 Maxwell's Equations Project 339 t 11 Numerical Integration and Monte Carlo Methods Numerical Integration Methods in One Dimension Simple Monte Carlo Evaluation of Integrals Numerical Integration of Multidimensional Integrals Monte Carlo Error Analysis Nonuniform Probability Distributions Neutron Transport Importance Sampling Metropolis Monte Carlo Method A Error Estimates for Numerical Integration B The Standard Deviation of the Mean 370 HC The Acceptance-Rejection Method Random Walks Introduction Modified Random Walks Applications to Polymers Diffusion Controlled Chemical Reactions The Continuum Limit Random Number Sequences Projects Percolation Introduction The Percolation Threshold Cluster Labeling Critical Exponents and Finite Size Scaling The Renormalization Group Projects Fractals The Fractal Dimension Regulär Fractals Fractal Growth Processes Fractals and Chaos Many Dimensions Projects 493
6 Contents XIII 15 Complexity Cellular Automata Lattice Gas Models of Fluid Flow Self-Organized Critical Phenomenon Neural Networks Genetic Algorithms Overview and Projects The Microcanonical Ensemble Introduction The Microcanonical Ensemble The Demon Algorithm One-Dimensional Classical Ideal Gas The Temperature and the Canonical Ensemble The Ising Model HeatFlow Comment A Relation of the Mean Demon Energy to the Temperature Monte Carlo Simulation of the Canonical Ensemble The Canonical Ensemble The Metropolis Algorithm Verification of the Boltzmann Distribution The Ising Model The Ising Phase Transition Other Applications of the Ising Model Simulation of Classical Fluids Optimized Monte Carlo Data Analysis Other Ensembles More Applications Projects A Fluctuations in the Canonical Ensemble B Exact Enumeration of the 2 x 2 Ising Model Quantum Systems Introduction Review of Quantum Theory Bound State Solutions The Time-Dependent Schrödinger Equation 635
7 xiv Contents 18.5 Introduction to Variational Methods Random Walk Quantum Monte Carlo Diffusion Quantum Monte Carlo Path Integral Quantum Monte Carlo Epilogue: The Same Algorithms Give the Same Results The Unity of Physics Percolation and Galaxies Numbers, Pretty Pictures, and Insight What are Computers Doing to Physics? 670 Appendixes A From BASIC to FORTRAN 673 B From BASIC to C 693 Index 715
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