Basic Concepts in Computational Physics

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Springer Science & Business Media, Dec 11, 2013 - Science - 377 pages
With the development of ever more powerful computers a new branch of physics and engineering evolved over the last few decades: Computer Simulation or Computational Physics. It serves two main purposes:
- Solution of complex mathematical problems such as, differential equations, minimization/optimization, or high-dimensional sums/integrals.
- Direct simulation of physical processes, as for instance, molecular dynamics or Monte-Carlo simulation of physical/chemical/technical processes.
Consequently, the book is divided into two main parts: Deterministic methods and stochastic methods. Based on concrete problems, the first part discusses numerical differentiation and integration, and the treatment of ordinary differential equations. This is augmented by notes on the numerics of partial differential equations. The second part discusses the generation of random numbers, summarizes the basics of stochastics which is then followed by the introduction of various Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. All this is again augmented by numerous applications from physics. The final two chapters on Data Analysis and Stochastic Optimization share the two main topics as a common denominator. The book offers a number of appendices to provide the reader with more detailed information on various topics discussed in the main part. Nevertheless, the reader should be familiar with the most important concepts of statistics and probability theory albeit two appendices have been dedicated to provide a rudimentary discussion.
 

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Contents

1 Some Basic Remarks
1
Part I Deterministic Methods
14
2 Numerical Differentiation
17
3 Numerical Integration
29
4 The KEPLER Problem
51
Initial Value Problems
60
6 The Double Pendulum
81
7 Molecular Dynamics
97
15 The ISING Model
209
16 Some Basics of Stochastic Processes
229
17 The Random Walk and Diffusion Theory
251
18 MarkovChain Monte Carlo and the Potts Model
275
19 Data Analysis
287
20 Stochastic Optimization
298
Appendix A The TwoBody Problem
315
The Newton Method
321

Boundary Value Problems
110
9 The OneDimensional Stationary Heat Equation
123
10 The OneDimensional Stationary Schrödinger Equation
131
11 Partial Differential Equations
147
Part II Stochastic Methods
169
12 Pseudo Random Number Generators
170
13 Random Sampling Methods
185
14 A Brief Introduction to MonteCarlo Methods
196
Appendix C Numerical Solution of Linear Systemsof Equations
323
Appendix D Basics of Probability Theory
331
Appendix E Phase Transitions
344
Appendix F Fractional Integrals and Derivatives in 1D
349
Appendix G Least Squares Fit
351
Appendix H Deterministic Optimization
357
Index
369
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About the author (2013)

Ewald Schachinger
Institut für Theoretische und Computational Physik,
Technische Universität Graz, Petersgasse 16, A-8010 Graz
schachinger@itp.tugraz.ac.at
Benjamin A. Stickler
Institut für Theoretische Physik, Karl Franzens Universität
Graz, Universitätsplatz 5, A-8010 Graz, benjamin.stickler@uni-graz.at

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