| Molecular Dynamics | |
| Introduction to Molecular Dynamics | p. 3 |
| Basic Approach | p. 3 |
| Macroscopic Parameters | p. 6 |
| Inter-Atomic Potentials | p. 8 |
| Numerical Integration Techniques | p. 14 |
| Analysis of MD Runs | p. 18 |
| From Classical to Quantum-Mechanical MD | p. 23 |
| Ab Initio MD | p. 24 |
| Car-Parrinello Molecular Dynamics | p. 25 |
| Potential Energy Surface | p. 28 |
| Advanced Numerical Methods | p. 29 |
| References | p. 37 |
| Wigner Function Quantum Molecular Dynamics | p. 41 |
| Quantum Distribution Functions | p. 41 |
| Semiclassical Molecular Dynamics | p. 43 |
| Quantum Dynamics | p. 50 |
| Time Correlation Functions in the Canonical Ensemble | p. 54 |
| Discussion | p. 58 |
| References | p. 59 |
| Classical Monte Carlo | |
| The Monte Carlo Method, an Introduction | p. 63 |
| What is a Monte Carlo Calculation? | p. 63 |
| Random Number Generation | p. 67 |
| Integration by Monte Carlo | p. 71 |
| Summary | p. 77 |
| References | p. 78 |
| Monte Carlo Methods in Classical Statistical Physics | p. 79 |
| Introduction | p. 79 |
| Statistical Physics Primer | p. 80 |
| The Monte Carlo Method | p. 85 |
| Cluster Algorithms | p. 93 |
| Statistical Analysis of Monte Carlo Data | p. 99 |
| Reweighting Techniques | p. 108 |
| Finite-Size Scaling Analysis | p. 114 |
| Generalized Ensemble Methods | p. 129 |
| Concluding Remarks | p. 135 |
| References | p. 135 |
| The Monte Carlo Method for Particle Transport Problems | p. 141 |
| Transport Problems and Stochastic Processes | p. 141 |
| The Transport Equation: Fredholm Integral Equation of Second Kind | p. 143 |
| The Boltzmann Equation | p. 144 |
| The Linear Integral Equation for the Collision Density | p. 147 |
| Monte Carlo Solution | p. 150 |
| Some Special Sampling Techniques | p. 154 |
| An Illustrative Example | p. 156 |
| References | p. 158 |
| Kinetic Modelling | |
| The Particle-in-Cell Method | p. 161 |
| General Remarks | p. 161 |
| Integration of Equations of Particle Motion | p. 163 |
| Plasma Source and Boundary Effects | p. 166 |
| Calculation of Plasma Parameters and Fields Acting on Particles | p. 170 |
| Solution of Maxwell's Equations | p. 175 |
| Particle Collisions | p. 183 |
| Final Remarks | p. 188 |
| References | p. 188 |
| Gyrokinetic and Gyrofluid Theory and Simulation of Magnetized Plasmas | p. 191 |
| Introduction | p. 191 |
| Single Particle Dynamics | p. 193 |
| Continuum Gyrokinetics | p. 200 |
| Gyrofluid Model | p. 204 |
| Gyrokinetic Particle Simulation Model | p. 207 |
| Gyrokinetic Particle Simulation Model Applications | p. 210 |
| Summary | p. 217 |
| References | p. 218 |
| Semiclassical Approaches | |
| Boltzmann Transport in Condensed Matter | p. 223 |
| Boltzmann Equation for Quasiparticles | p. 223 |
| Techniques for the Solution of the Boltzmann Equation | p. 230 |
| Conclusions | p. 252 |
| References | p. 253 |
| Semiclassical Description of Quantum Many-Particle Dynamics in Strong Laser Fields | p. 255 |
| Semiclassical Many-Particle Dynamics in Mean-Field Approximation | p. 255 |
| Semiclassical Ground State | p. 261 |
| Application to Simple-Metal Clusters | p. 265 |
| References | p. 272 |
| Quantum Monte Carlo | |
| World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons | p. 277 |
| Introduction | p. 277 |
| Discrete Imaginary Time World Lines for the XXZ Spin Chain | p. 278 |
| World-Line Representations without Discretization Error | p. 299 |
| Loop Operator Representation of the Heisenberg Model | p. 303 |
| Spin-Phonon Simulations | p. 308 |
| Auxiliary Field Quantum Monte Carlo Methods | p. 312 |
| Numerical Stabilization Schemes for Lattice Models | p. 325 |
| The Hirsch-Fye Impurity Algorithm | p. 337 |
| Selected Applications of the Auxiliary Field Method | p. 344 |
| Conclusion | p. 345 |
| The Trotter Decomposition | p. 345 |
| The Hubbard-Stratonovich Decomposition | p. 347 |
| Slater Determinants and their Properties | p. 349 |
| References | p. 353 |
| Autocorrelations in Quantum Monte Carlo Simulations of Electron-Phonon Models | p. 357 |
| Introduction | p. 357 |
| Holstein Model | p. 358 |
| Numerical Methods | p. 358 |
| Problem of Autocorrelations | p. 360 |
| Origin of Autocorrelations and Principal Components | p. 363 |
| Conclusions | p. 365 |
| References | p. 366 |
| Diagrammatic Monte Carlo and Stochastic Optimization Methods for Complex Composite Objects in Macroscopic Baths | p. 367 |
| Introduction | p. 367 |
| Physical Properties of Interest | p. 372 |
| The Diagrammatic Monte Carlo Method | p. 374 |
| Stochastic Optimization Method | p. 391 |
| Conclusions and Perspectives | p. 393 |
| References | p. 394 |
| Path Integral Monte Carlo Simulation of Charged Particles in Traps | p. 397 |
| Introduction | p. 397 |
| Idea of Path Integral Monte Carlo | p. 397 |
| Basic Numerical Issues of PIMC | p. 401 |
| PIMC for Degenerate Bose Systems | p. 406 |
| Discussion | p. 410 |
| References | p. 411 |
| Ab-Initio Methods in Physics and Chemistry | |
| Ab-Initio Approach to the Many-Electron Problem | p. 415 |
| Introduction | p. 415 |
| An Orbital Approach to Chemistry | p. 419 |
| Hartree-Fock Theory | p. 427 |
| Density Functional Theory | p. 432 |
| References | p. 435 |
| Ab-Initio Methods Applied to Structure Optimization and Microscopic Modelling | p. 437 |
| Exploring Energy Hypersurfaces | p. 437 |
| Applied Theoretical Chemistry | p. 444 |
| Model Hamiltonians | p. 451 |
| Summary and Outlook | p. 465 |
| Links to Popular Ab Initio Packages | p. 466 |
| References | p. 467 |
| Effective Field Approaches | |
| Dynamical Mean-Field Approximation and Cluster Methods for Correlated Electron Systems | p. 473 |
| Introduction | p. 473 |
| Mean-Field Theory for Correlated Electron Systems | p. 475 |
| Extending the DMFT: Effective Cluster Theories | p. 492 |
| Conclusions | p. 499 |
| References | p. 501 |
| Local Distribution Approach | p. 505 |
| Introduction | p. 505 |
| Applications of the LD Approach | p. 514 |
| Summary | p. 525 |
| References | p. 526 |
| Iterative Methods for Sparse Eigenvalue Problems | |
| Exact Diagonalization Techniques | p. 529 |
| Basis Construction | p. 529 |
| Eigenstates of Sparse Matrices | p. 539 |
| References | p. 543 |
| Chebyshev Expansion Techniques | p. 545 |
| Chebyshev Expansion and Kernel Polynomial Approximation | p. 545 |
| Applications of the Kernel Polynomial Method | p. 554 |
| KPM in Relation to other Numerical Approaches | p. 568 |
| References | p. 575 |
| The Density Matrix Renormalisation Group: Concepts and Applications | |
| The Conceptual Background of Density-Matrix Renormalization | p. 581 |
| Introduction | p. 581 |
| Entangled States | p. 581 |
| Reduced Density Matrices | p. 582 |
| Solvable Models | p. 583 |
| Spectra | p. 586 |
| Entanglement Entropy | p. 589 |
| Matrix-Product States | p. 593 |
| Summary | p. 594 |
| References | p. 594 |
| Density-Matrix Renormalization Group Algorithms | p. 597 |
| Introduction | p. 597 |
| Matrix-Product States and (Super-)Blocks | p. 598 |
| Numerical Renormalization Group | p. 600 |
| Infinite-System DMRG Algorithm | p. 602 |
| Finite-System DMRG Algorithm | p. 607 |
| Additive Quantum Numbers | p. 611 |
| Truncation Errors | p. 613 |
| Computational Cost and Optimization | p. 616 |
| Basic Extensions | p. 617 |
| References | p. 618 |
| Dynamical Density-Matrix Renormalization Group | p. 621 |
| Introduction | p. 621 |
| Methods for Simple Discrete Spectra | p. 623 |
| Dynamical DMRG | p. 626 |
| Finite-Size Scaling | p. 630 |
| Momentum-Dependent Quantities | p. 631 |
| Application: Spectral Function of the Hubbard Model | p. 632 |
| References | p. 634 |
| Studying Time-Dependent Quantum Phenomena with the Density-Matrix Renormalization Group | p. 637 |
| Time Dependence in Interacting Quantum Systems | p. 637 |
| Sudden Quench of Interacting Fermions | p. 643 |
| Discussion | p. 650 |
| References | p. 651 |
| Applications of Quantum Information in the Density-Matrix Renormalization Group | p. 653 |
| Basic Concepts of Quantum Information Theory | p. 653 |
| Entropic Analysis of Quantum Phase Transitions | p. 657 |
| Discussion and Outlook | p. 662 |
| References | p. 663 |
| Density-Matrix Renormalization Group for Transfer Matrices: Static and Dynamical Properties of 1D Quantum Systems at Finite Temperature | p. 665 |
| Introduction | p. 665 |
| Quantum Transfer Matrix Theory | p. 666 |
| The Method - DMRG Algorithm for the QTM | p. 669 |
| An Example: The Spin-1/2 Heisenberg Chain with Staggered and Uniform Magnetic Fields | p. 671 |
| Impurity and Boundary Contributions | p. 672 |
| Real-Time Dynamics | p. 673 |
| References | p. 676 |
| Concepts of High Performance Computing | |
| Architecture and Performance Characteristics of Modern High Performance Computers | p. 681 |
| Microprocessors | p. 682 |
| Parallel Computing | p. 701 |
| Conclusion and Outlook | p. 729 |
| References | p. 729 |
| Optimization Techniques for Modern High Performance Computers | p. 731 |
| Optimizing Serial Code | p. 732 |
| Shared-Memory Parallelization | p. 755 |
| Conclusion and Outlook | p. 766 |
| References | p. 767 |
| Appendix: Abbreviations | p. 769 |
| Index | p. 773 |
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