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Computational physics: simulation of classical and quantum systems

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Bibliographische Detailangaben
1. Verfasser:	Scherer, Philipp O. J. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cham [u.a.] Springer 2013
Ausgabe:	2. ed.
Schriftenreihe:	Graduate texts in physics
Schlagworte:	Computerphysik Lehrbuch
Online-Zugang:	Klappentext Inhaltsverzeichnis
Beschreibung:	XVIII, 454 S. Ill., graph. Darst.
ISBN:	9783319004006

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Datensatz im Suchindex

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adam_text	Contents Part I Numerical Methods 1 Error Analysis .............................. 3 1 .1 Machine Numbers and Rounding Errors .............. 3 1.2 Numerical Errors of Elementary Floating Point Operations ..... 6 1.2.1 Numerical Extinction .................... 7 1.2.2 Addition ........................... 8 1.2.3 Multiplication ........................ 9 1.3 Error Propagation .......................... 9 1.4 Stability of Iterative Algorithms ................... 11 1.5 Example: Rotation .......................... 12 1.6 Truncation Error ........................... 13 1.7 Problems ............................... 14 2 Interpolation ............................... 15 2.1 Interpolating Functions ....................... 15 2.2 Polynomial Interpolation ...................... 16 2.2.1 Lagrange Polynomials .................... 17 2.2.2 В ary centric Lagrange Interpolation ............. 17 2.2.3 Newton s Divided Differences ................ 18 2.2.4 Neville Method ....................... 20 2.2.5 Error of Polynomial Interpolation .............. 21 2.3 Spline Interpolation ......................... 22 2.4 Rational Interpolation ........................ 25 2.4.1 Padé Approximant ...................... 25 2.4.2 Barycentric Rational Interpolation ............. 27 2.5 Multivariate Interpolation ...................... 32 2.6 Problems ............................... 33 3 Numerical Differentiation ........................ 37 3.1 One-Sided Difference Quotient ................... 37 3.2 Central Difference Quotient ..................... 38 xi xii Contents 3.3 Extrapolation Methods........................ 39 3.4 Higher Derivatives.......................... 41 3.5 Partial Derivatives of Multivariate Functions ............ 42 3.6 Problems............................... 43 4 Numerical Integration.......................... 45 4.1 Equidistant Sample Points...................... 46 4.1.1 Closed Newton-Cotes Formulae ............... 46 4.1.2 Open Newton-Cotes Formulae ............... 48 4.1.3 Composite Newton-Cotes Rules ............... 48 4.1.4 Extrapolation Method (Romberg Integration) ........ 49 4.2 Optimized Sample Points ...................... 50 4.2.1 Clenshaw-Curtis Expressions ................ 50 4.2.2 Gaussian Integration ..................... 52 4.3 Problems ............................... 56 5 Systems of Inhomogeneous Linear Equations ............. 59 5.1 Gaussian Elimination Method .................... 60 5.1.1 Pivoting ........................... 63 5.1.2 Direct LU Decomposition .................. 63 5.2 QR Decomposition ......................... 64 5.2.1 QR Decomposition by Orthogonalization .......... 64 5.2.2 QR Decomposition by Householder Reflections ...... 66 5.3 Linear Equations with Tridiagonal Matrix .............. 69 5.4 Cyclic Tridiagonal Systems ..................... 71 5.5 Iterative Solution of Inhomogeneous Linear Equations ....... 73 5.5.1 General Relaxation Method ................. 73 5.5.2 Jacobi Method ........................ 73 5.5.3 Gauss-Seidel Method .................... 74 5.5.4 Damping and Successive Over-Relaxation ......... 75 5.6 Conjugate Gradients ......................... 76 5.7 Matrix Inversion ........................... 77 5.8 Problems ............................... 78 6 Roots and Extremal Points ........................ 83 6.1 Root Finding ............................. 83 6.1.1 Bisection ........................... 84 6.1.2 Regula Falsi (False Position) Method ............ 85 6.1.3 Newton-Raphson Method .................. 85 6.1.4 Secant Method ........................ 86 6.1.5 Interpolation ......................... 87 6.1.6 Inverse Interpolation ..................... 88 6.1.7 Combined Methods ..................... 91 6.1.8 Multidimensional Root Finding ............... 97 6.1.9 Quasi-Newton Methods ................... 98 Contents xiii 6.2 Function Minimization ....................... 99 6.2.1 The Ternary Search Method ................. 99 6.2.2 The Golden Section Search Method (Brent s Method) ... 101 6.2.3 Minimization in Multidimensions .............. 106 6.2.4 Steepest Descent Method .................. 106 6.2.5 Conjugate Gradient Method ................. 107 6.2.6 Newton-Raphson Method .................. 107 6.2.7 Quasi-Newton Methods ................... 108 6.3 Problems ............................... 110 Fourier Transformation ........ 7.1 Fourier Integral and Fourier Series 7.2 Discrete Fourier Transformation . . 7.2.1 Trigonometric Interpolation 7.2.2 Real Valued Functions 13 13 14 16 18 7.2.3 Approximate Continuous Fourier Transformation ..... 119 7.3 Fourier Transform Algorithms .................... 120 7.3.1 Goertzel s Algorithm .................... 120 7.3.2 Fast Fourier Transformation ................. 121 7.4 Problems ............................... 125 8 Random Numbers and Monte Carlo Methods ............. 127 8.1 Some Basic Statistics ........................ 127 8.1.1 Probability Density and Cumulative Probability Distribution 127 8.1.2 Histogram .......................... 128 8.1.3 Expectation Values and Moments .............. 129 8.1.4 Example: Fair Die ...................... 130 8Л.5 Normal Distribution ..................... 131 8.1.6 Multivariate Distributions .................. 132 8.1.7 Central Limit Theorem ................... 133 8.1.8 Example: Binomial Distribution ............... 133 8.1.9 Average of Repeated Measurements ............. 134 8.2 Random Numbers .......................... 135 8.2.1 Linear Congruent Mapping ................. 135 8.2.2 Marsaglia-Zamann Method ................. 135 8.2.3 Random Numbers with Given Distribution ......... 136 8.2.4 Examples ........................... 136 8.3 Monte Carlo Integration ....................... 138 8.3.1 Numerical Calculation of π ................. 138 8.3.2 Calculation of an Integral .................. 139 8.3.3 More General Random Numbers .............. 140 8.4 Monte Carlo Method for Thermodynamic Averages ........ 141 8.4.1 Simple Sampling ....................... 141 8.4.2 Importance Sampling .................... 142 8.4.3 Metropolis Algorithm .................... 142 8.5 Problems ............................... 144 xiv Contents 9 Eigenvalue Problems ........................... 147 9.1 Direct Solution ............................ 148 9.2 Jacobi Method ............................ 148 9.3 Tridiagonal Matrices ......................... 150 9.3.1 Characteristic Polynomial of a Tridiagonal Matrix ..... 151 9.3.2 Special Tridiagonal Matrices ................ 151 9.3.3 The QL Algorithm ..................... 156 9.4 Reduction to a Tridiagonal Matrix .................. 157 9.5 Large Matrices ............................ 159 9.6 Problems ............................... 160 10 Data Fitting ................................ 161 10.1 Least Square Fit ........................... 162 10.1.1 Linear Least Square Fit ................... 163 10.1.2 Linear Least Square Fit with Orthogonalization ...... 165 10.2 Singular Value Decomposition ................... 167 10.2.1 Full Singular Value Decomposition ............. 168 10.2.2 Reduced Singular Value Decomposition .......... 168 10.2.3 Low Rank Matrix Approximation .............. 170 10.2.4 Linear Least Square Fit with Singular Value Decomposition 172 10.3 Problems ............................... 175 11 Discretization of Differential Equations ................ 177 11.1 Classification of Differential Equations ............... 178 11.1.1 Linear Second Order PDE .................. 178 11.1.2 Conservation Laws ..................... 179 11.2 Finite Differences .......................... 180 11.2.1 Finite Differences in Time .................. 181 11.2.2 Stability Analysis ...................... 182 11.2.3 Method of Lines ....................... 183 11.2.4 Eigenvector Expansion ................... 183 11.3 Finite Volumes ............................ 185 11.3.1 Discretization of fluxes ................... 188 11.4 Weighted Residual Based Methods ................. 190 11.4.1 Point Collocation Method .................. 191 .4.2 Sub-domain Method .....................191 .4.3 Least Squares Method ....................192 .4.4 Galerkin Method .......................192 11.5 Spectral and Pseudo-spectral Methods ............... 193 11.5.1 Fourier Pseudo-spectral Methods .............. 193 11.5.2 Example: Polynomial Approximation ............ 194 11.6 Finite Elements .......................... . 196 .6.1 One-Dimensional Elements .................196 .6.2 Two- and Three-Dimensional Elements ...........197 .6.3 One-Dimensional Galerkin FEM ..............201 11.7 Boundary Element Method .....................204 Contents xv 12 Equations of Motion ...........................207 12.1 The State Vector .......................... 208 12.2 Time Evolution of the State Vector ................ 209 12.3 Explicit Forward Euler Method .................. 210 12.4 Implicit Backward Euler Method ................. 212 12.5 Improved Euler Methods ...................... 213 12.6 Taylor Series Methods ....................... 215 12.6.1 Nordsieck Predictor-Corrector Method ..........215 12.6.2 Gear Predictor-Corrector Methods ............217 12.7 Runge- Kutta Methods .......................217 12.7.1 Second Order Runge-Kutta Method ........... 218 12.7.2 Third Order Runge-Kutta Method ............ 218 12.7.3 Fourth Order Runge-Kutta Method ............ 219 12.8 Quality Control and Adaptive Step Size Control ......... 220 12.9 Extrapolation Methods ....................... 221 12.10 Linear Multistep Methods ..................... 222 12.10.1 Adams-Bashforth Methods ................222 12.10.2 Adams-Moulton Methods .................223 12.10.3 Backward Differentiation (Gear) Methods ........223 12.10.4 Predictor-Corrector Methods ...............224 12.11 Verlet Methods ...........................225 12.11.1 Liouville Equation .................... 225 12.11.2 Split-Operator Approximation .............. 226 12.11.3 Position Verlet Method .................. 227 12.11.4 Velocity Verlet Method .................. 227 12.11.5 Stornier- Verlet Method .................. 228 12.11.6 Error Accumulation for the Störmer-Verlet Method . . . 229 12.11.7 Beeman s Method ..................... 230 12.11.8 The Leapfrog Method ................... 231 12.12 Problems .............................. 232 Part II Simulation of Classical and Quantum Systems 13 Rotational Motion ............................239 13.1 Transformation to a Body Fixed Coordinate System .......239 13.2 Properties of the Rotation Matrix .................240 13.3 Properties of W, Connection with the Vector of Angular Velocity 242 13.4 Transformation Properties of the Angular Velocity ........244 13.5 Momentum and Angular Momentum ...............246 13.6 Equations of Motion of a Rigid Body ...............246 13.7 Moments of Inertia .........................247 13.8 Equations of Motion for a Rotor ..................248 13.9 Explicit Methods ..........................248 13.10 Loss of Orthogonality .......................250 13.11 Implicit Method ..........................251 13.12 Kinetic Energy of a Rotor .....................255 xvi Contents 13.13 Parametrization by Euler Angles .................. 255 13.14 Cayley-Klein Parameters, Quaternions, Euler Parameters ..... 256 13.15 Solving the Equations of Motion with Quaternions ........ 259 13.16 Problems .............................. 260 14 Molecular Mechanics ...........................263 14.1 Atomic Coordinates ........................264 14.2 Force Fields ............................266 14.2.1 Intramolecular Forces ................... 267 14.2.2 Intermolecular Interactions ................ 269 14.3 Gradients .............................. 270 14.4 Normal Mode Analysis ....................... 274 14.4.1 Harmonic Approximation ................. 274 14.5 Problems .............................. 276 15 Thermodynamic Systems ........................279 15.1 Simulation of a Lennard-Jones Fluid ...............279 15.1.1 Integration of the Equations of Motion .......... 280 15.1.2 Boundary Conditions and Average Pressure ....... 281 15.1.3 Initial Conditions and Average Temperature ....... 281 15.1.4 Analysis of the Results .................. 282 15.2 Monte Carlo Simulation ...................... 287 15.2.1 One-Dimensional Ising Model ..............287 15.2.2 Two-Dimensional Ising Model ..............289 15.3 Problems ..............................290 16 Random Walk and Brownian Motion ..................293 16.1 Markovian Discrete Time Models ................. 293 16.2 Random Walk in One Dimension ................. 294 16.2.1 Random Walk with Constant Step Size .......... 295 16.3 The Freely Jointed Chain ...................... 296 16.3.1 Basic Statistic Properties ................. 297 16.3.2 Gyration Tensor ...................... 299 16.3.3 Hookean Spring Model .................. 300 16.4 Langevin Dynamics ........................ 301 16.5 Problems .............................. 303 17 Electrostatics ...............................305 17.1 Poisson Equation ..........................305 17.1.1 Homogeneous Dielectric Medium ............ 306 17.1.2 Numerical Methods for the Poisson Equation ...... 307 17.1.3 Charged Sphere ...................... 309 17.1.4 Variables ......................... 311 17.1.5 Discontinuous ε ...................... 313 17.1.6 Sol vation Energy of a Charged Sphere .......... 314 17.1.7 The Shifted Grid Method ................. 314 Contents xvii 17.2 Poisson-Boltzmann Equation.................... 315 17.2.1 Linearization of the Poisson-BoItzmann Equation ..... 317 17.2.2 Discretization of the Linearized Poisson-Boltzmann Equation ........................... 318 17.3 Boundary Element Method for the Poisson Equation ........ 318 17.3.1 Integral Equations for the Potential ............. 318 17.3.2 Calculation of the Boundary Potential ............ 321 17.4 Boundary Element Method for the Linearized Poisson-Boltzmann Equation ............................... 324 17.5 Electrostatic Interaction Energy (Onsager Model) ......... 325 17.5.1 Example: Point Charge in a Spherical Cavity ........ 326 17.6 Problems ............................... 327 18 Waves ................................... 329 18.1 Classical Waves ........................... 329 18.2 Spatial Discretization in One Dimension .............. 332 18.3 Solution by an Eigenvector Expansion ............... 334 18.4 Discretization of Space and Time .................. 337 18.5 Numerical Integration with a Two-Step Method .......... 338 18.6 Reduction to a First Order Differential Equation .......... 340 18.7 Two-Variable Method ........................ 343 18.7.1 Leapfrog Scheme ...................... 343 18.7.2 Lax-Wendroff Scheme .................... 345 18.7.3 Crank-Nicolson Scheme ................... 347 18.8 Problems ............................... 349 19 Diffusion .................................. 351 19.1 Particle Flux and Concentration Changes .............. 351 19.2 Diffusion in One Dimension ..................... 353 19.2.1 Explicit Euler (Forward Time Centered Space) Scheme . . 353 19.2.2 Implicit Euler (Backward Time Centered Space) Scheme . 355 19.2.3 Crank-Nicolson Method ................... 357 19.2.4 Error Order Analysis ..................... 358 19.2.5 Finite Element Discretization ................ 360 19.3 Split-Operator Method for Multidimensions ............ 360 19.4 Problems ............................... 362 20 Nonlinear Systems ............................ 363 20.1 Iterated Functions .......................... 364 20.1.1 Fixed Points and Stability .................. 364 20.1.2 The Lyapunov Exponent ................... 366 20.1.3 The Logistic Map ...................... 367 20.1.4 Fixed Points of the Logistic Map .............. 367 20.1.5 Bifurcation Diagram ..................... 369 20.2 Population Dynamics ........................ 370 20.2.1 Equilibria and Stability ................... 370 20.2.2 The Continuous Logistic Model ............... 371 xviii Contents 20.3 Lotka- Volteira Model........................ 372 20.3.1 Stability Analysis ...................... 372 20.4 Functional Response ......................... 373 20.4.1 Holling-Tanner Model .................... 375 20.5 Reaction-Diffusion Systems ..................... 378 20.5.1 General Properties of Reaction-Diffusion Systems ..... 378 20.5.2 Chemical Reactions ..................... 378 20.5.3 Diffusive Population Dynamics ............... 379 20.5.4 Stability Analysis ...................... 379 20.5.5 Lotka- Volterra Model with Diffusion ............ 380 20.6 Problems ............................... 382 21 Simple Quantum Systems ........................385 21.1 Pure and Mixed Quantum States ...................386 21.1.1 Wavefunctions ........................387 21.1.2 Density Matrix for an Ensemble of Systems ........387 21.1.3 Time Evolution of the Density Matrix ............388 21.2 Wave Packet Motion in One Dimension ...............389 21.2.1 Discretization of the Kinetic Energy ............ 390 21.2.2 Time Evolution ....................... 392 21.2.3 Example: Free Wave Packet Motion ............. 402 21.3 Few-State Systems .......................... 403 21.3.1 Two-State System ...................... 405 21.3.2 Two-State System with Time Dependent Perturbation . . . 408 21.3.3 Superexchange Model .................... 410 21.3.4 Ladder Model for Exponential Decay ............ 412 21.3.5 Landau-Zener Model .................... 414 21.4 The Dissipative Two-State System ................. 416 21.4.1 Equations of Motion for a Two-State System ........416 21.4.2 The Vector Model ......................417 21.4.3 TheSpin-i System .....................418 21 Λ A Relaxation Processes — The Bloch Equations ........420 21.4.5 The Driven Two-State System ................421 21.4.6 Elementary Qubit Manipulation ...............428 21.5 Problems ...............................430 Appendix I Performing the Computer Experiments ........... 433 Appendix II Methods and Algorithms ................... 435 References ................................... 441 Index ...................................... 449 Graduate Texts in Physics Philipp OJ. Scherer Computational Physics Simulation of Classical and Quantum Systems, Second Edition This textbook presents basic and advanced computational physics in a very didactic style. It con¬ tains very-well-presented and simple mathematical descriptions of many of the most important algorithms used in computational physics. Many cîear mathematical descriptions of important techniques in computational physics are given. The first part of the hook discusses the basic numerical methods. A large number of exercises and computer experiments allows to study the properties of these methods. In addition to polynomial interpolation by the methods of Lagrange and Newton baryeentric rational interpolation is explained. Besides the classical root finding methods, inverse interpolation is discussed, together with the popular combined methods by Пеккег and Brent and a not so well known improvement by Chandrupatla. A general crmpter on the numerical treatment of differential equations provides methods of finite differences, finite volumes, finite elements and boundary elements together with spectral methods and weighted residual based methods. The book uses a rather general concept for the equation of motion which can be applied to ordinary and partial differential equations. Several classes of integration methods are discussed including not only the standard Kuler and Runge Kulta method but also multislep methods and the class of Verlet methods which is introduced by studying the motion in I.iouville space. A comparison of several methods for quantum wavepacket motion is performed, containing pseudo-spectral methods, finite differences methods, rational approximation to the time evolution operator, second order differencing and split operator methods, lhe second part concentrates on simulation of classical and quantum systems. The book gives simple but non trivial examples from a broad range of physical topucs trying to give the reader insight into the numerical treatment but also the simulated problems. Rotational motion is treated in much detail to describe the motion of rigid rotors which can be just a simple spinning top or a collection of molecules or planets. Ihe behaviour of simple quantum systems is studied thoroughly. One focus is on a two level system in an external field. Solution of the Bloch equations allows the simulation of a quantum bit and lo understand elementary principles from quantum optics. As an example of a ihermodynamic system, the Lennard-Jones fluid is simulated. Ihe principles of molecular dynamics are shown vith practical simulations. A second thermodynamic topic is the Ising model in one and two dimensions. The solution of the Pois¬ son Boltzman equation is discussed in detail which is very important in Biophysics as well as in semiconductor physics. Besides the standard finite differences methods, also modern boundary element methods are discussed. Wave?, and diffusion processes are simulated. Différent methods are compared with regard to their stability and efficiency. Random walk models are studied with application to basic polymer physics. Nonlinear systems are discussed in detail with application to population dynamics and reaction diffusion systems. Ihe exercises to the book are realized as computer experiments. A large number of Java applets is provided. It can be tried out by the reader even without programming skills. Ihe interested reader can modify the programs with the help of the freely available and platform independent programming environment netbeans . Physics ► springerxom MATERIALS extras.spring er. com
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oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-026103299
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spelling	Scherer, Philipp O. J. Verfasser (DE-588)110745108 aut Computational physics simulation of classical and quantum systems Philipp O. J. Scherer 2. ed. Cham [u.a.] Springer 2013 XVIII, 454 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in physics Computerphysik (DE-588)4273564-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Computerphysik (DE-588)4273564-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-00401-3 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026103299&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026103299&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Scherer, Philipp O. J. Computational physics simulation of classical and quantum systems Computerphysik (DE-588)4273564-6 gnd
subject_GND	(DE-588)4273564-6 (DE-588)4123623-3
title	Computational physics simulation of classical and quantum systems
title_auth	Computational physics simulation of classical and quantum systems
title_exact_search	Computational physics simulation of classical and quantum systems
title_full	Computational physics simulation of classical and quantum systems Philipp O. J. Scherer
title_fullStr	Computational physics simulation of classical and quantum systems Philipp O. J. Scherer
title_full_unstemmed	Computational physics simulation of classical and quantum systems Philipp O. J. Scherer
title_short	Computational physics
title_sort	computational physics simulation of classical and quantum systems
title_sub	simulation of classical and quantum systems
topic	Computerphysik (DE-588)4273564-6 gnd
topic_facet	Computerphysik Lehrbuch
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026103299&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026103299&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
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