Basic Concepts in Computational Physics

✍ Scribed by Stickler, Benjamin A;Schachinger, Ewald

• Publisher: Springer
• Year: 2016
• Tongue: English
• Leaves: 409
• Edition: 2nd ed. 2016
• Category: Library

✦ Synopsis

This new edition is a concise introduction to the basic methods of computational physics. Readers will discover the benefits of numerical methods for solving complex mathematical problems and for the direct simulation of physical processes.

The book is divided into two main parts: Deterministic methods and stochastic methods in computational physics. Based on concrete problems, the first part discusses numerical differentiation and integration, as well as the treatment of ordinary differential equations. This is extended by a brief introduction to the numerics of partial differential equations. The second part deals with the generation of random numbers, summarizes the basics of stochastics, and subsequently introduces Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. The final two chapters discuss data analysis and stochastic optimization. All this is again motivated and augmented by applications from physics. In addition, the book offers a number of appendices to provide the reader with information on topics not discussed in the main text.

Numerous problems with worked-out solutions, chapter introductions and summaries, together with a clear and application-oriented style support the reader. Ready to use C++ codes are provided online.

✦ Table of Contents

Preface......Page 5
Acknowledgments......Page 8
Contents......Page 9
1.1 Motivation......Page 15
1.2 Roundoff Errors......Page 20
1.3 Methodological Errors......Page 21
1.4 Stability......Page 23
1.5 Concluding Remarks......Page 26
References......Page 27
Part I Deterministic Methods......Page 28
2.1 Introduction......Page 29
2.2 Finite Differences......Page 30
2.3 Finite Difference Derivatives......Page 32
2.4 A Systematic Approach: The Operator Technique......Page 35
2.5 Concluding Discussion......Page 38
Problems......Page 41
References......Page 42
3.1 Introduction......Page 43
3.2 Rectangular Rule......Page 44
3.3 Trapezoidal Rule......Page 47
3.4 The Simpson Rule......Page 49
3.5 General Formulation: The Newton-Cotes Rules......Page 50
3.6 Gauss-Legendre Quadrature......Page 53
3.7 An Example......Page 58
3.8 Concluding Discussion......Page 60
Problems......Page 62
References......Page 64
4.1 Introduction......Page 65
4.2 Numerical Treatment......Page 68
Problems......Page 72
References......Page 73
5.1 Introduction......Page 74
5.2 Simple Integrators......Page 76
Taylor Series Methods......Page 77
Linear Multi-step Methods......Page 78
5.3 Runge-Kutta Methods......Page 79
Implicit Midpoint:......Page 82
5.4 Hamiltonian Systems: Symplectic Integrators......Page 84
Symplectic Euler......Page 86
5.5 An Example: The Kepler Problem, Revisited......Page 87
Symplectic Euler......Page 88
Summary......Page 92
References......Page 93
6.1 Hamilton's Equations......Page 95
6.2 Numerical Solution......Page 99
6.3 Numerical Analysis of Chaos......Page 104
Summary......Page 109
Problems......Page 110
References......Page 111
7.1 Introduction......Page 112
7.2 Classical Molecular Dynamics......Page 113
7.3 Numerical Implementation......Page 117
Boundary Conditions......Page 118
Initialization and Equilibration......Page 120
Summary......Page 122
Problems......Page 123
References......Page 125
8.1 Introduction......Page 126
8.2 Finite Difference Approach......Page 128
8.3 Shooting Methods......Page 133
Summary......Page 137
References......Page 138
9.1 Introduction......Page 139
9.2 Finite Differences......Page 140
9.3 A Second Scenario......Page 142
Problems......Page 145
References......Page 146
10.1 Introduction......Page 147
10.2 A Simple Example: The Particle in a Box......Page 151
10.3 Numerical Solution......Page 155
10.4 Another Case......Page 159
References......Page 163
11.1 Introduction......Page 165
11.2 The Poisson Equation......Page 166
11.3 The Time-Dependent Heat Equation......Page 171
11.4 The Wave Equation......Page 175
11.5 The Time-Dependent Schrödinger Equation......Page 178
Summary......Page 186
References......Page 187
Part II Stochastic Methods......Page 189
12.1 Introduction......Page 190
12.2 Different Methods......Page 193
Linear Congruential Generators......Page 194
Fibonacci Generators......Page 195
Statistical Tests......Page 197
Hypothesis Testing......Page 198
Problems......Page 201
References......Page 202
13.1 Introduction......Page 203
13.2 Inverse Transformation Method......Page 206
13.3 Rejection Method......Page 208
13.4 Probability Mixing......Page 212
Problems......Page 214
References......Page 215
14.1 Introduction......Page 216
14.2 Monte-Carlo Integration......Page 218
14.3 The Metropolis Algorithm: An Introduction......Page 224
Summary......Page 227
References......Page 228
15.1 The Model......Page 229
15.2 Numerics......Page 240
(1) Lattice Geometry......Page 241
(2) Initialization......Page 242
(3) Execution of the Code......Page 243
15.3 Selected Results......Page 244
Problems......Page 249
References......Page 250
16.1 Introduction......Page 251
16.2 Stochastic Processes......Page 252
16.3 Markov Processes......Page 255
16.4 Markov-Chains......Page 263
16.5 Continuous-Time Markov-Chains......Page 270
Summary......Page 272
References......Page 273
17.1 Introduction......Page 275
Basics......Page 277
Moments......Page 279
Recurrence......Page 281
17.3 The Wiener Process and Brownian Motion......Page 283
17.4 Generalized Diffusion Models......Page 289
Summary......Page 297
References......Page 298
18.1 Introduction......Page 300
18.2 Markov-Chain Monte Carlo Methods......Page 301
18.3 The Potts Model......Page 305
18.4 Advanced Algorithms for the Potts Model......Page 309
Summary......Page 311
References......Page 312
19.2 Calculation of Errors......Page 314
19.3 Auto-Correlations......Page 318
19.4 The Histogram Technique......Page 322
Summary......Page 323
References......Page 324
20.1 Introduction......Page 325
20.2 Hill Climbing......Page 327
20.3 Simulated Annealing......Page 329
(i) Proposal Probability......Page 330
(iii) Cooling Strategy......Page 331
20.4 Genetic Algorithms......Page 336
20.5 Some Further Methods......Page 338
Summary......Page 339
References......Page 340
A The Two-Body Problem......Page 342
B Solving Non-linear Equations: The Newton Method......Page 348
C Numerical Solution of Linear Systems of Equations......Page 350
C.1 The LU Decomposition......Page 351
C.2 The Gauss-Seidel Method......Page 354
D Fast Fourier Transform......Page 358
E.1 Classical Definition......Page 363
E.2 Random Variables and Moments......Page 364
E.3 Binomial Distribution and Limit Theorems......Page 366
E.4 Poisson Distribution and Counting Experiments......Page 367
E.5 Continuous Variables......Page 368
E.6 Bayes' Theorem......Page 369
E.8 Central Limit Theorem......Page 370
E.10 The Correlation Coefficient......Page 371
E.11 Stable Distributions......Page 373
F.1 Some Basics......Page 375
F.2 Landau Theory......Page 376
G Fractional Integrals and Derivatives in 1D......Page 379
H.1 Motivation......Page 381
H.2 Linear Least Squares Fit......Page 383
H.3 Nonlinear Least Squares Fit......Page 384
I.1 Introduction......Page 387
I.2 Steepest Descent......Page 388
I.3 Conjugate Gradients......Page 390
References......Page 399
Index......Page 401