Computational Physics

Random Number Generation.- 1 Introduction.- 2 The Miracle Number 16807.- 3 Bit Strings of Kirkpatrick-Stoll.- 4 A Modern Example.- 5 Problems.- 6 Summary.- References.- A Few Exercises with Random Numbers.- Monte Carlo Simulations of Spin Systems.- 1 Introduction.- 2 Spin Models and Phase Transitions.- 2.1 Models and Observables.- 2.2 Phase Transitions.- 3 The Monte Carlo Method.- 3.1 Estimators and Autocorrelation Times.- 3.2 Metropolis Algorithm.- 3.3 Cluster Algorithms.- 3.4 Multicanonical Algorithms for First-Order Transitions.- 4 Reweighting Techniques.- 5 Applications to the 3D Heisenberg Model.- 5.1 Simulations for T>Tc.- 5.2 Simulations near Tc.- 6 Concluding Remarks.- Appendix: Program Codes.- References.- Metastable Systems and Stochastic Optimization.- 1 An Introduction to Complex Systems.- 2 Dynamics in Complex Systems.- 2.1 Thermal Relaxation Dynamics: The Metropolis Algorithm.- 2.2 Thermal Relaxation Dynamics: A Marcov Process.- 2.3 Thermal Relaxation Dynamics: A Simple Example.- 3 Modeling Constant-Temperature Thermal Relaxation.- 3.1 Coarse-Graining a Complex State Space.- 3.2 Tree Dynamics.- 3.3 A Serious Application: Aging Effects in Spin Glasses.- 4 Stochastic Optimization: How to Find the Ground State of Complex Systems.- 4.1 Simulated Annealing.- 4.2 Optimal Simulated Annealing Schedules: A Simple Example.- 4.3 Adaptive Annealing Schedules and the Ensemble Approach to Simulated Annealing.- 5 Summary.- Appendix: Examples and Exercises (with S. Schubert).- References.- Modelling and Computer Simulation of Granular Media.- 1 The Physics of Granular Media.- 1.1 What are Granular Media?.- 1.2 Stress Distribution in Granular Packing: Arching.- 1.3 Dilatancy, Fluidization and Collisional Cooling.- 1.4 Stick-and-Slip Motion and Self-Organized Criticality (with S. Dippel).- 1.5 Segregation, Convection, Heaping (with S. Dippel).- 2 Molecular Dynamics Simulations I: Soft Particles.- 2.1 General Remarks.- 2.2 Normal Force.- 2.3 Tangential Force.- 2.4 Detachment Effect.- 2.5 Brake Failure Effect (with J. Schäfer).- 3 Molecular Dynamic Simulations II: Hard Particles (with J. Schäfer).- 3.1 Event-Driven Simulation.- 3.2 Collision Operator.- 3.3 Limitations.- 4 Contact Dynamics Simulations (with L. Brendel and F. Radjai).- 4.1 General Remarks.- 4.2 Contact Laws and Equations of Motion.- 4.3 Iterative Determination of Forces and Accelerations.- 4.4 Results.- 5 The Bottom-to-Top Restructuring Model.- 5.1 The Algorithm and its Justification (with E. Jobs).- 5.2 Simulation of a Rotating Drum (with T. Scheffler and G. Baumann).- 6 Conclusion.- References.- Algorithms for Biological Aging.- 1 Introduction.- 2 Concepts and Models.- 3 Techniques.- 4 Results.- References.- Simulations of Chemical Reactions.- 1 Introduction.- 2 The Basic Kinetic Approach.- 3 Numerical and Analytical Approaches for Reactions Under Diffusion.- 4 Reactions in Layered Systems.- 5 Reactions Under Mixing.- 6 Reactions Controlled by Enhanced Diffusion.- References.- Random Walks on Fractals.- 1 Introduction.- 2 Deterministic Fractals.- 2.1 The Koch Curve.- 2.2 The Sierpinski Gasket.- 3 Random Fractals.- 3.1 The Random-Walk Trail.- 3.2 Self-Avoiding Walks.- 3.3 Percolation.- 4 The "Chemical Distance" ?.- 5 Random Walks on Fractals.- 5.1 Root Mean Square Displacement R(t).- 5.2 The Mean Probability Density.- 6 Biased Diffusion.- 7 Numerical Approaches.- 7.1 Generation of Percolation Clusters.- 7.2 Simulation of Random Walks.- 8 Description of the Programs.- References.- Multifractal Characteristics of Electronic Wave Functions in Disordered Systems.- 1 Electronic States in Disordered Systems.- 2 The Anderson Model of Localization.- 3 Calculation of the Eigenvectors.- 4 Description of Multifractal Objects.- 5 Multifractal Analysis of the Wave Functions.- 6 Computation of the Multifractal Characteristics.- 7 Topical Results of the Multifractal Analysis.- References.- Transfer-Matrix Methods and Finite-Size Scaling for Disordered Systems.- 1 Introduction.- 2 One-Dimensional Systems.- 2.1 The Transfer Matrix.- 2.2 The Ordered Limit.- 2.3 The Localization Length.- 2.4 Resolvent Method.- 3 Finite-Size Scaling.- 4 Numerical Evaluation of the Anderson Transition.- 4.1 Localization Length of Quasi-1D Systems.- 4.2 Dependence of the Localization Length on the Cross Section.- 4.3 Finite-Size Scaling Numerically.- 5 Present Status of the Results from Transfer-Matrix Calculations 185 References.- Quantum Monte Carlo Investigations for the Hubbard Model.- 1 Introduction.- 1.1 The Hubbard Model.- 1.2 What to Compute.- 1.3 Quantum Simulations.- 2 Grand Canonical Quantum Monte Carlo.- 2.1 The Trotter-Suzuki Transformation.- 2.2 The Hubbard-Stratonovich Transformation.- 2.3 The Partition Function.- 2.4 The Monte Carlo Weight.- 3 Equal-Time Greens Functions.- 3.1 Single Spin Updates.- 3.2 Numerical Instabilities.- 4 History and Further Reading.- Appendix A: Statistical Monte Carlo Methods.- Appendix B: OCTAVE.- Appendix C: Exercises.- References.- Quantum Dynamics in Nanoscale Devices.- 1 Introduction.- 2 Theory.- 3 Data Analysis.- 4 Implementation.- 5 Application: Quantum Interference of Two Identical Particles.- References.- Quantum Chaos.- 1 Classical and Quantum Chaos.- 2 Quantum Time Evolution.- 3 Quantum State Tomography.- 3.1 Phase-Space Distributions.- 3.2 Phase-Space Entropy.- 4 Case Study: A Driven Anharmonic Quantum Oscillator.- 4.1 Classical Phase-Space Dynamics.- 4.2 Quantum Phase-Space Dynamics.- 4.3 Quasienergy Spectra.- 4.4 Chaotic Tunneling.- 5 Concluding Remarks.- References.- Numerical Simulation in Quantum Field Theory.- 1 Quantum Field Theory and Particle Physics.- 1.1 Particles, Fields, Standard Model.- 1.2 Beyond Perturbation Theory.- 2 Lattice Formulation of Field Theory.- 2.1 Path Integral.- 2.2 Lattice Regularization.- 2.3 Field Theory and Critical Phenomena.- 2.4 Effective Field Theory.- 3 Stochastic Evaluation of Path Integrals.- 3.1 Monte Carlo Method.- 3.2 Metropolis Algorithm for ?4.- 4 Summary.- Appendix: FORTRAN Monte Carlo Package for ?4.- References.- Modeling and a Simulation Method for Molecular Systems.- 1 Introduction.- 2 Brief Review of the Simulation Method.- 3 Modeling of Polymer Systems.- 4 Coarse-Graining.- 5 The Monomer Unit.- 6 Bonded Interactions for BPA-PC.- 7 Parallelization of the Polymer System.- References.- Constraints in Molecular Dynamics, Nonequilibrium Processes in Fluids via Computer Simulations.- 1 Introduction.- 2 Basics of Molecular Dynamics.- 2.1 Equations of Motion.- 2.2 Extraction of Data from MD Simulations.- 3 Potentials, Constraints, and Integrators.- 3.1 Interaction Potential and Scaling.- 3.2 Thermostats.- 3.3 Integrators.- 4 Nonequilibrium Phenomena.- 4.1 Relaxation Processes.- 4.2 Plane Couette Flow.- 4.3 Viscosity.- 4.4 Structural Changes.- 4.5 Colloidal Dispersions.- 4.6 Mixtures.- 5 Complex Fluids.- 5.1 Polymer Melts.- 5.2 Nematic Liquid Crystals.- 5.3 Ferrofluids and Magneto-Rheological Fluids.- References.- Molecular-Dynamic Simulations of Structure Formation in Complex Materials.- 1 Introduction.- 2 Simulation Methods.- 3 Total Energies and Interatomic Forces.- 3.1 Classical Concepts.- 3.2 Density-Functional Theory, Car-Parrinello MD.- 4 Density-Functional Based Tight-Binding Method.- 4.1 Creation of the Pseudoatoms.- 4.2 Calculation of Matrix Elements.- 4.3 Fitting of Short-Range Repulsive Part.- 5 Vibrational Properties.- 6 Simulation Geometries and Regimes.- 6.1 Clusters, Molecules.- 6.2 Bulk-Crystalline and Amorphous Solids.- 6.3 Surfaces and Adsorbates.- 7 Accuracy and Transferability.- 7.1 Small Silicon Clusters, Sin.- 7.2 Molecules, Hydrocarbons.- 7.3 Solid Crystalline Modifications, Silicon.- 8 Applications.- 8.1 Structure and Stability of Polymerized C60.- 8.2 Stability of Highly Tetrahedral Amorphous Carbon, ta-C.- 8.3 Diamond Surface Reconstructions.- 9 Summary.- References.- Finite Element Methods for the Stokes Equation.- 1 Introduction.- 2 Stokes Equation.- 2.1 Conservation Equations.- 2.2 Function Spaces and Variational Formulation.- 2.3 Saddle Point Problem.- 2.4 General Boundary Conditions.- 2.5 Example.- 3 Discretization.- 3.1 General Formulation.- 3.2 Finite Elements for Saddle-Point Problems.- 4 Final Remarks.- References.- Principles of Parallel Computers and Some Impacts on Their Programming Models.- 1 Introduction.- 2 Overview on Architecture Principles.- 3 General Classification.- 4 Multiprocessor Systems.- 5 Massively Parallel Processor Systems.- 6 Multiple Shared-Memory Multiprocessors.- 7 Multithreading Programming Model.- 8 Message-Passing Programming Model.- 9 Summary.- References.- Parallel Programming Styles: A Brief Overview.- 1 Introduction.- 2 Programming Models.- 2.1 Definition.- 2.2 Classification.- 3 Programming a Shared Memory Computer.- 3.1 The KSR Programming Model.- 3.2 Levels of Parallelism.- 3.3 Program Implementation.- 3.4 Examples.- 4 Programming a Distributed Memory Computer Using PARIX.- 4.1 What is PARIX.- 4.2 PARIX Hardware Environment.- 4.3 Communication and Process Model Under PARIX.- 4.4 Programming Model.- 4.5 An Example, PARIX says "Hello World".- 5 Programming Heterogenous Workstation Clusters Using MPI.- 5.1 Introduction.- 5.2 Basic Structure of MPICH.- 5.3 What Is Included in MPI?.- 5.4 What Does the Standard Exclude?.- 5.5 MPI Says "Hello World".- 5.6 Current Available Implementations of MPI.- 6 Summary.- References.