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Basic Concepts in Computational Physics 2014 ed. [Kõva köide]

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  • Formaat: Hardback, 377 pages, kõrgus x laius x paksus: 234x156x22 mm, kaal: 730 g, black & white illustrations
  • Ilmumisaeg: 11-Dec-2013
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319024345
  • ISBN-13: 9783319024349
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Basic Concepts in Computational Physics 2014 ed.Suurem pilt
  • Formaat: Hardback, 377 pages, kõrgus x laius x paksus: 234x156x22 mm, kaal: 730 g, black & white illustrations
  • Ilmumisaeg: 11-Dec-2013
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319024345
  • ISBN-13: 9783319024349
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319024349.html
Märksõnad:

This book presents both deterministic methods and stochastic methods with numerous applications from physics. Coverage includes numerical differentiation and integration, monte-carlo (MC) methods, data analysis, and stochastic optimization.



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.

Arvustused

From the reviews: "The authors characterize the aim of their book to 'address the scenarios of direct simulation of physical processes and the solution of complex mathematical problems on a very basic level'. It is directed to lecturers teaching basic courses in Computational Physics and to students as a companion when starting studying in this field." (Rolf Dieter Grigorieff, zbMATH, Vol. 1287, 2014)

1 Some Basic Remarks
1(16)
1.1 Motivation
1(5)
1.2 Rounding Errors
6(2)
1.3 Methodological Errors
8(1)
1.4 Stability
9(3)
1.5 Concluding Remarks
12(1)
References
13(4)
Part I Deterministic Methods
2 Numerical Differentiation
17(12)
2.1 Introduction
17(1)
2.2 Finite Differences
18(2)
2.3 Finite Difference Derivatives
20(2)
2.4 A Systematic Approach: The Operator Technique
22(3)
2.5 Concluding Discussion
25(2)
Summary
27(1)
Problems
28(1)
References
28(1)
3 Numerical Integration
29(22)
3.1 Introduction
29(1)
3.2 Rectangular Rule
30(3)
3.3 Trapezoidal Rule
33(2)
3.4 The Simpson Rule
35(1)
3.5 General Formulation: The Newton-Cotes Rules
36(2)
3.6 Gauss-Legendre Quadrature
38(6)
3.7 An Example
44(1)
3.8 Concluding Discussion
45(2)
Summary
47(1)
Problems
48(1)
References
49(2)
4 The Kepler Problem
51(10)
4.1 Introduction
51(1)
4.2 The Problem
52(2)
4.3 Numerical Treatment
54(5)
Summary
59(1)
References
59(2)
5 Ordinary Differential Equations: Initial Value Problems
61(20)
5.1 Introduction
61(1)
5.2 Simple Integrators
62(4)
5.3 Runge-Kutta Methods
66(5)
5.4 Hamiltonian Systems: Symplectic Integrators
71(2)
5.5 An Example: The Kepler Problem, Revisited
73(5)
Summary
78(1)
Problems
79(1)
References
79(2)
6 The Double Pendulum
81(16)
6.1 Hamilton's Equations
81(4)
6.2 Numerical Solution
85(3)
6.3 Numerical Analysis of Chaos
88(7)
Summary
95(1)
Problems
95(1)
References
96(1)
7 Molecular Dynamics
97(14)
7.1 Introduction
97(1)
7.2 Classical Molecular Dynamics
97(5)
7.3 Numerical Implementation
102(5)
Summary
107(1)
Problems
108(1)
References
109(2)
8 Numerics of Ordinary Differential Equations: Boundary Value Problems
111(12)
8.1 Introduction
111(2)
8.2 Finite Difference Approach
113(5)
8.3 Shooting Methods
118(3)
Summary
121(1)
References
122(1)
9 The One-Dimensional Stationary Heat Equation
123(8)
9.1 Introduction
123(1)
9.2 Finite Differences
124(2)
9.3 A Second Scenario
126(2)
Summary
128(1)
Problems
129(1)
Reference
129(2)
10 The One-Dimensional Stationary Schrodinger Equation
131(16)
10.1 Introduction
131(3)
10.2 A Simple Example: The Particle in a Box
134(5)
10.3 Numerical Solution
139(3)
10.4 Another Case
142(3)
Summary
145(1)
Problems
146(1)
References
146(1)
11 Partial Differential Equations
147(24)
11.1 Introduction
147(1)
11.2 The Poisson Equation
148(3)
11.3 The Time-Dependent Heat Equation
151(6)
11.4 The Wave Equation
157(3)
11.5 The Time-Dependent Schrodinger Equation
160(7)
Summary
167(1)
Problems
167(1)
References
168(3)
Part II Stochastic Methods
12 Pseudo Random Number Generators
171(14)
12.1 Introduction
171(3)
12.2 Different Approaches
174(4)
12.3 Quality Tests
178(4)
Summary
182(1)
Problems
182(1)
References
183(2)
13 Random Sampling Methods
185(12)
13.1 Introduction
185(2)
13.2 Inverse Transformation Method
187(3)
13.3 Rejection Method
190(3)
13.4 Probability Mixing
193(2)
Summary
195(1)
Problems
195(2)
14 A Brief Introduction to Monte-Carlo Methods
197(12)
14.1 Introduction
197(1)
14.2 Monte-Carlo Integration
198(7)
14.3 The Metropolis Algorithm: An Introduction
205(3)
Summary
208(1)
References
208(1)
15 The Ising Model
209(20)
15.1 The Model
209(9)
15.2 Numerics
218(5)
15.3 Selected Results
223(4)
Summary
227(1)
Problems
228(1)
References
228(1)
16 Some Basics of Stochastic Processes
229(22)
16.1 Introduction
229(1)
16.2 Stochastic Processes
230(3)
16.3 Markov Processes
233(8)
16.4 MARKov-Chains
241(6)
16.5 Continuous-Time MARKov-Chains
247(2)
Summary
249(1)
Problems
250(1)
References
250(1)
17 The Random Walk and Diffusion Theory
251(24)
17.1 Introduction
251(2)
17.2 The Random Walk
253(6)
17.3 The Wiener Process and Brownian Motion
259(6)
17.4 Generalized Diffusion Models
265(7)
Summary
272(1)
Problems
272(1)
References
273(2)
18 Markov-Chain Monte Carlo and the Potts Model
275(12)
18.1 Introduction
275(1)
18.2 Markov-Chain Monte Carlo Methods
276(3)
18.3 The Potts Model
279(5)
18.4 Advanced Algorithms for the Potts Model
284(1)
Summary
285(1)
Problems
286(1)
References
286(1)
19 Data Analysis
287(12)
19.1 Introduction
287(1)
19.2 Calculation of Errors
287(4)
19.3 Auto-Correlations
291(4)
19.4 The Histogram Technique
295(1)
Summary
296(1)
Problems
297(1)
Reference
297(2)
20 Stochastic Optimization
299(16)
20.1 Introduction
299(2)
20.2 Hill Climbing
301(2)
20.3 Simulated Annealing
303(7)
20.4 Genetic Algorithms
310(2)
20.5 Some Further Methods
312(1)
Summary
313(1)
Problems
314(1)
References
314(1)
Appendix A The Two-Body Problem 315(6)
Appendix B Solving Non-Linear Equations: The Newton Method 321(2)
Appendix C Numerical Solution of Linear Systems of Equations 323(8)
Appendix D Basics of Probability Theory 331(14)
Appendix E Phase Transitions 345(4)
Appendix F Fractional Integrals and Derivatives in 1D 349(2)
Appendix G Least Squares Fit 351(6)
Appendix H Deterministic Optimization 357(12)
Index 369
Ewald Schachinger Institut fur Theoretische und Computational Physik, Technische Universitat Graz, Petersgasse 16, A-8010 Graz schachinger@itp.tugraz.ac.at Benjamin A. Stickler Institut fur Theoretische Physik, Karl Franzens Universitat Graz, Universitatsplatz 5, A-8010 Graz, benjamin.stickler@uni-graz.at