Computational Physics. Simulation of Classical and Quantum Systems 2nd ed.

Computational Physics......Page 4
 Preface to the Second Edition......Page 7
 Preface to the First Edition......Page 9
 Contents......Page 11
Part I: Numerical Methods......Page 19
  1.1 Machine Numbers and Rounding Errors......Page 20
  1.2 Numerical Errors of Elementary Floating Point Operations......Page 23
   1.2.1 Numerical Extinction......Page 24
   1.2.2 Addition......Page 25
  1.3 Error Propagation......Page 26
  1.4 Stability of Iterative Algorithms......Page 28
  1.5 Example: Rotation......Page 29
  1.6 Truncation Error......Page 30
  1.7 Problems......Page 31
  2.1 Interpolating Functions......Page 32
  2.2 Polynomial Interpolation......Page 33
   2.2.2 Barycentric Lagrange Interpolation......Page 34
   2.2.3 Newton's Divided Differences......Page 35
   2.2.4 Neville Method......Page 37
   2.2.5 Error of Polynomial Interpolation......Page 38
  2.3 Spline Interpolation......Page 39
   2.4.1 Padé Approximant......Page 42
   2.4.2 Barycentric Rational Interpolation......Page 44
    2.4.2.1 Rational Interpolation of Order [M,N]......Page 45
    2.4.2.2 Rational Interpolation Without Poles......Page 48
  2.5 Multivariate Interpolation......Page 49
  2.6 Problems......Page 50
  3.1 One-Sided Difference Quotient......Page 53
  3.2 Central Difference Quotient......Page 54
  3.3 Extrapolation Methods......Page 55
  3.4 Higher Derivatives......Page 57
  3.5 Partial Derivatives of Multivariate Functions......Page 58
  3.6 Problems......Page 59
 Chapter 4: Numerical Integration......Page 60
   4.1.1 Closed Newton-Cotes Formulae......Page 61
   4.1.3 Composite Newton-Cotes Rules......Page 63
   4.1.4 Extrapolation Method (Romberg Integration)......Page 64
   4.2.1 Clenshaw-Curtis Expressions......Page 65
   4.2.2 Gaussian Integration......Page 67
    4.2.2.1 Gauss-Legendre Integration......Page 68
    4.2.2.3 Connection with an Eigenvalue Problem......Page 70
  4.3 Problems......Page 71
 Chapter 5: Systems of Inhomogeneous Linear Equations......Page 73
  5.1 Gaussian Elimination Method......Page 74
   5.1.2 Direct LU Decomposition......Page 77
   5.2.1 QR Decomposition by Orthogonalization......Page 78
   5.2.2 QR Decomposition by Householder Reflections......Page 80
  5.3 Linear Equations with Tridiagonal Matrix......Page 83
  5.4 Cyclic Tridiagonal Systems......Page 85
   5.5.2 Jacobi Method......Page 87
   5.5.3 Gauss-Seidel Method......Page 88
   5.5.4 Damping and Successive Over-Relaxation......Page 89
  5.6 Conjugate Gradients......Page 90
  5.7 Matrix Inversion......Page 91
  5.8 Problems......Page 92
  6.1 Root Finding......Page 96
   6.1.1 Bisection......Page 97
   6.1.3 Newton-Raphson Method......Page 98
   6.1.4 Secant Method......Page 99
   6.1.5 Interpolation......Page 100
   6.1.6 Inverse Interpolation......Page 101
    6.1.7.1 Dekker's Method......Page 104
    6.1.7.2 Brent's Method......Page 105
    6.1.7.3 Chandrupatla's method......Page 108
   6.1.8 Multidimensional Root Finding......Page 110
   6.1.9 Quasi-Newton Methods......Page 111
   6.2.1 The Ternary Search Method......Page 112
   6.2.2 The Golden Section Search Method (Brent's Method)......Page 114
   6.2.4 Steepest Descent Method......Page 119
   6.2.6 Newton-Raphson Method......Page 120
   6.2.7 Quasi-Newton Methods......Page 121
  6.3 Problems......Page 123
  7.1 Fourier Integral and Fourier Series......Page 125
  7.2 Discrete Fourier Transformation......Page 126
   7.2.1 Trigonometric Interpolation......Page 128
   7.2.2 Real Valued Functions......Page 130
   7.2.3 Approximate Continuous Fourier Transformation......Page 131
   7.3.1 Goertzel's Algorithm......Page 132
   7.3.2 Fast Fourier Transformation......Page 133
  7.4 Problems......Page 137
   8.1.1 Probability Density and Cumulative Probability Distribution......Page 139
   8.1.2 Histogram......Page 140
   8.1.3 Expectation Values and Moments......Page 141
   8.1.4 Example: Fair Die......Page 142
   8.1.5 Normal Distribution......Page 143
   8.1.6 Multivariate Distributions......Page 144
   8.1.8 Example: Binomial Distribution......Page 145
   8.1.9 Average of Repeated Measurements......Page 146
   8.2.2 Marsaglia-Zamann Method......Page 147
    8.2.4.1 Fair Die......Page 148
    8.2.4.3 Random Points on the Unit Sphere......Page 149
   8.3.1 Numerical Calculation of pi......Page 150
   8.3.2 Calculation of an Integral......Page 151
   8.3.3 More General Random Numbers......Page 152
   8.4.1 Simple Sampling......Page 153
   8.4.3 Metropolis Algorithm......Page 154
  8.5 Problems......Page 156
 Chapter 9: Eigenvalue Problems......Page 158
  9.2 Jacobi Method......Page 159
  9.3 Tridiagonal Matrices......Page 161
    9.3.2.1 Discretized Second Derivatives......Page 162
    9.3.2.2 Discretized First Derivatives......Page 166
   9.3.3 The QL Algorithm......Page 167
  9.4 Reduction to a Tridiagonal Matrix......Page 168
  9.5 Large Matrices......Page 170
  9.6 Problems......Page 171
 Chapter 10: Data Fitting......Page 172
  10.1 Least Square Fit......Page 173
   10.1.1 Linear Least Square Fit......Page 174
   10.1.2 Linear Least Square Fit with Orthogonalization......Page 176
  10.2 Singular Value Decomposition......Page 178
   10.2.2 Reduced Singular Value Decomposition......Page 179
   10.2.3 Low Rank Matrix Approximation......Page 181
   10.2.4 Linear Least Square Fit with Singular Value Decomposition......Page 183
  10.3 Problems......Page 186
 Chapter 11: Discretization of Differential Equations......Page 187
   11.1.1 Linear Second Order PDE......Page 188
   11.1.2 Conservation Laws......Page 189
  11.2 Finite Differences......Page 190
   11.2.1 Finite Differences in Time......Page 191
   11.2.2 Stability Analysis......Page 192
   11.2.4 Eigenvector Expansion......Page 193
  11.3 Finite Volumes......Page 195
   11.3.1 Discretization of fluxes......Page 198
  11.4 Weighted Residual Based Methods......Page 200
   11.4.2 Sub-domain Method......Page 201
   11.4.4 Galerkin Method......Page 202
   11.5.1 Fourier Pseudo-spectral Methods......Page 203
    11.5.2.2 Sub-domain Method......Page 204
    11.5.2.3 Galerkin Method......Page 205
   11.6.1 One-Dimensional Elements......Page 206
    11.6.2.1 Triangulation......Page 207
    11.6.2.2 Rectangular Elements......Page 209
   11.6.3 One-Dimensional Galerkin FEM......Page 211
  11.7 Boundary Element Method......Page 214
 Chapter 12: Equations of Motion......Page 216
  12.1 The State Vector......Page 217
  12.2 Time Evolution of the State Vector......Page 218
  12.3 Explicit Forward Euler Method......Page 219
  12.4 Implicit Backward Euler Method......Page 221
  12.5 Improved Euler Methods......Page 222
   12.6.1 Nordsieck Predictor-Corrector Method......Page 224
  12.7 Runge-Kutta Methods......Page 226
   12.7.2 Third Order Runge-Kutta Method......Page 227
   12.7.3 Fourth Order Runge-Kutta Method......Page 228
  12.8 Quality Control and Adaptive Step Size Control......Page 229
  12.9 Extrapolation Methods......Page 230
   12.10.1 Adams-Bashforth Methods......Page 231
   12.10.3 Backward Differentiation (Gear) Methods......Page 232
   12.10.4 Predictor-Corrector Methods......Page 233
   12.11.1 Liouville Equation......Page 234
   12.11.2 Split-Operator Approximation......Page 235
   12.11.4 Velocity Verlet Method......Page 236
   12.11.5 Störmer-Verlet Method......Page 237
   12.11.6 Error Accumulation for the Störmer-Verlet Method......Page 238
   12.11.7 Beeman's Method......Page 239
   12.11.8 The Leapfrog Method......Page 240
  12.12 Problems......Page 241
Part II: Simulation of Classical and Quantum Systems......Page 245
  13.1 Transformation to a Body Fixed Coordinate System......Page 246
  13.2 Properties of the Rotation Matrix......Page 247
  13.3 Properties of W, Connection with the Vector of Angular Velocity......Page 249
  13.4 Transformation Properties of the Angular Velocity......Page 251
  13.6 Equations of Motion of a Rigid Body......Page 253
  13.7 Moments of Inertia......Page 254
  13.9 Explicit Methods......Page 255
  13.10 Loss of Orthogonality......Page 257
  13.11 Implicit Method......Page 258
  13.13 Parametrization by Euler Angles......Page 262
  13.14 Cayley-Klein Parameters, Quaternions, Euler Parameters......Page 263
  13.15 Solving the Equations of Motion with Quaternions......Page 266
  13.16 Problems......Page 267
 Chapter 14: Molecular Mechanics......Page 270
  14.1 Atomic Coordinates......Page 271
  14.2 Force Fields......Page 273
   14.2.1 Intramolecular Forces......Page 274
   14.2.2 Intermolecular Interactions......Page 276
  14.3 Gradients......Page 277
   14.4.1 Harmonic Approximation......Page 281
  14.5 Problems......Page 283
  15.1 Simulation of a Lennard-Jones Fluid......Page 285
   15.1.1 Integration of the Equations of Motion......Page 286
   15.1.3 Initial Conditions and Average Temperature......Page 287
   15.1.4 Analysis of the Results......Page 288
    15.1.4.1 Deviation from the Ideal Gas Behavior......Page 289
    15.1.4.2 Structural Order......Page 290
    15.1.4.3 Ballistic and Diffusive Motion......Page 291
   15.2.1 One-Dimensional Ising Model......Page 293
   15.2.2 Two-Dimensional Ising Model......Page 295
  15.3 Problems......Page 296
  16.1 Markovian Discrete Time Models......Page 298
  16.2 Random Walk in One Dimension......Page 299
   16.2.1 Random Walk with Constant Step Size......Page 300
  16.3 The Freely Jointed Chain......Page 301
   16.3.1 Basic Statistic Properties......Page 302
   16.3.2 Gyration Tensor......Page 304
   16.3.3 Hookean Spring Model......Page 305
  16.4 Langevin Dynamics......Page 306
  16.5 Problems......Page 308
  17.1 Poisson Equation......Page 310
   17.1.1 Homogeneous Dielectric Medium......Page 311
   17.1.2 Numerical Methods for the Poisson Equation......Page 312
   17.1.3 Charged Sphere......Page 314
   17.1.4 Variable epsilon......Page 315
   17.1.5 Discontinuous epsilon......Page 318
   17.1.7 The Shifted Grid Method......Page 319
  17.2 Poisson-Boltzmann Equation......Page 320
   17.2.1 Linearization of the Poisson-Boltzmann Equation......Page 322
   17.3.1 Integral Equations for the Potential......Page 323
   17.3.2 Calculation of the Boundary Potential......Page 326
  17.4 Boundary Element Method for the Linearized Poisson-Boltzmann Equation......Page 329
  17.5 Electrostatic Interaction Energy (Onsager Model)......Page 330
   17.5.1 Example: Point Charge in a Spherical Cavity......Page 331
  17.6 Problems......Page 332
  18.1 Classical Waves......Page 334
  18.2 Spatial Discretization in One Dimension......Page 337
  18.3 Solution by an Eigenvector Expansion......Page 339
  18.4 Discretization of Space and Time......Page 342
  18.5 Numerical Integration with a Two-Step Method......Page 343
  18.6 Reduction to a First Order Differential Equation......Page 345
   18.7.1 Leapfrog Scheme......Page 348
   18.7.2 Lax-Wendroff Scheme......Page 350
   18.7.3 Crank-Nicolson Scheme......Page 352
  18.8 Problems......Page 354
  19.1 Particle Flux and Concentration Changes......Page 355
   19.2.1 Explicit Euler (Forward Time Centered Space) Scheme......Page 357
   19.2.2 Implicit Euler (Backward Time Centered Space) Scheme......Page 359
   19.2.3 Crank-Nicolson Method......Page 361
   19.2.4 Error Order Analysis......Page 362
  19.3 Split-Operator Method for Multidimensions......Page 364
  19.4 Problems......Page 366
 Chapter 20: Nonlinear Systems......Page 367
   20.1.1 Fixed Points and Stability......Page 368
   20.1.2 The Lyapunov Exponent......Page 370
   20.1.4 Fixed Points of the Logistic Map......Page 371
   20.1.5 Bifurcation Diagram......Page 373
   20.2.1 Equilibria and Stability......Page 374
   20.2.2 The Continuous Logistic Model......Page 375
   20.3.1 Stability Analysis......Page 376
  20.4 Functional Response......Page 377
   20.4.1 Holling-Tanner Model......Page 379
   20.5.2 Chemical Reactions......Page 382
   20.5.4 Stability Analysis......Page 383
   20.5.5 Lotka-Volterra Model with Diffusion......Page 384
  20.6 Problems......Page 386
 Chapter 21: Simple Quantum Systems......Page 388
  21.1 Pure and Mixed Quantum States......Page 389
   21.1.2 Density Matrix for an Ensemble of Systems......Page 390
   21.1.3 Time Evolution of the Density Matrix......Page 391
  21.2 Wave Packet Motion in One Dimension......Page 392
   21.2.1 Discretization of the Kinetic Energy......Page 393
    21.2.1.2 Finite Difference Methods......Page 394
    21.2.2.1 Rational Approximation......Page 395
    21.2.2.2 Second Order Differencing......Page 399
    21.2.2.3 Split-Operator Methods......Page 402
     Real-Space Product Formulae......Page 403
   21.2.3 Example: Free Wave Packet Motion......Page 405
  21.3 Few-State Systems......Page 406
   21.3.1 Two-State System......Page 408
   21.3.2 Two-State System with Time Dependent Perturbation......Page 411
   21.3.3 Superexchange Model......Page 413
   21.3.4 Ladder Model for Exponential Decay......Page 415
   21.3.5 Landau-Zener Model......Page 417
   21.4.1 Equations of Motion for a Two-State System......Page 419
   21.4.2 The Vector Model......Page 420
   21.4.3 The Spin-1/2 System......Page 421
    21.4.4.1 Phenomenological Description......Page 423
   21.4.5 The Driven Two-State System......Page 424
    21.4.5.1 Free Precession......Page 425
    21.4.5.2 Stationary Solution for Monochromatic Excitation......Page 426
    21.4.5.3 Excitation by a Resonant Pulse......Page 428
   21.4.6 Elementary Qubit Manipulation......Page 431
    21.4.6.1 Pauli-Gates......Page 432
  21.5 Problems......Page 433
Appendix I: Performing the Computer Experiments......Page 436
Appendix II: Methods and Algorithms......Page 438
References......Page 443
Index......Page 451