Computational Physics (Second Edition) by Thijssen J.
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Computational Physics (Second Edition) written by
Thijssen J. , Kavli Institute of Nanoscience, Delft University of Technology.
First published in 2007, this second edition describes the computational methods used in theoretical physics. New sections were added to cover finite element methods and lattice Boltzmann simulation, density functional theory, quantum molecular dynamics, Monte Carlo simulation, and diagonalisation of one-dimensional quantum systems. It covers many different areas of physics research and different computational methodologies, including computational methods such as Monte Carlo and molecular dynamics, various electronic structure methodologies, methods for solving partial differential equations, and lattice gauge theory. Throughout the book the relations between the methods used in different fields of physics are emphasised. Several new programs are described and can be downloaded from www.cambridge.org/9781107677135. The book requires a background in elementary programming, numerical analysis, and field theory, as well as undergraduate knowledge of condensed matter theory and statistical physics. It will be of interest to graduate students and researchers in theoretical, computational and experimental physics
Computational Physics (Second Edition) written by
Thijssen J.
cover the following topics.
Preface to the second edition
1. Introduction
1.1 Physics and computational physics
1.2 Classical mechanics and statistical mechanics
1.3 Stochastic simulations
1.4 Electrodynamics and hydrodynamics
1.5 Quantum mechanics
1.6 Relations between quantum mechanics and classical statistical physics
1.7 Quantum molecular dynamics
1.8 Quantum field theory
1.9 About this book
Exercises
References
2. Quantum scattering with a spherically symmetric potential
2.1 Introduction
2.2 A program for calculating cross sections
2.3 Calculation of scattering cross sections
Exercises
References
3. The variational method for the Schrödinger equation
3.1 Variational calculus
3.2 Examples of variational calculations
3.3 Solution of the generalised eigenvalue problem
3.4 Perturbation theory and variational calculus
Exercises
References
4. The Hartree–Fock method
4.1 Introduction
4.2 The Born–Oppenheimer approximation and the independent-particle method
4.3 The helium atom
4.4 Many-electron systems and the Slater determinant
4.5 Self-consistency and exchange: Hartree–Fock theory
4.6 Basis functions
4.7 The structure of a Hartree–Fock computer program
4.8 Integrals involving Gaussian functions
4.9 Applications and results
4.10 Improving upon the Hartree–Fock approximation
Exercises
References
5. Density functional theory
5.1 Introduction
5.2 The local density approximation
5.3 Exchange and correlation: a closer look
5.4 Beyond DFT: one- and two-particle excitations
5.5 A density functional program for the helium atom
5.6 Applications and results
Exercises
References
6. Solving the Schrödinger equation in periodic solids
6.1 Introduction: definitions
6.2 Band structures and Bloch’s theorem
6.3 Approximations
6.4 Band structure methods and basis functions
6.5 Augmented plane wave methods
6.6 The linearised APW (LAPW) method
6.7 The pseudopotential method
6.8 Extracting information from band structures
6.9 Some additional remarks
6.10 Other band methods
Exercises
References
7. Classical equilibrium statistical mechanics
7.1 Basic theory
7.2 Examples of statistical models; phase transitions
7.3 Phase transitions 184
7.4 Determination of averages in simulations
Exercises
References
8. Molecular dynamics simulations
8.1 Introduction
8.2 Molecular dynamics at constant energy
8.3 A molecular dynamics simulation program for argon
8.4 Integration methods: symplectic integrators
8.5 Molecular dynamics methods for different ensembles
8.6 Molecular systems
8.7 Long-range interactions
8.8 Langevin dynamics simulation
8.9 Dynamical quantities: nonequilibrium molecular dynamics
Exercises
References
9. Quantum molecular dynamics
9.1 Introduction
9.2 The molecular dynamics method
9.3 An example: quantum molecular dynamics for the hydrogen molecule
9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques
9.5 Implementation of the Car–Parrinello technique for pseudopotential DFT
Exercises
References
10. The Monte Carlo method
10.1 Introduction
10.2 Monte Carlo integration
10.3 Importance sampling through Markov chains
10.4 Other ensembles
10.5 Estimation of free energy and chemical potential
10.6 Further applications and Monte Carlo methods
10.7 The temperature of a finite system
Exercises
References
11. Transfer matrix and diagonalisation of spin chains
11.1 Introduction
11.2 The one-dimensional Ising model and the transfer matrix
11.3 Two-dimensional spin models
11.4 More complicated models
11.5 ‘Exact’ diagonalisation of quantum chains
11.6 Quantum renormalisation in real space
11.7 The density matrix renormalisation group method
Exercises
References
12. Quantum Monte Carlo methods
12.1 Introduction
12.2 The variational Monte Carlo method
12.3 Diffusion Monte Carlo
12.4 Path-integral Monte Carlo
12.5 Quantum Monte Carlo on a lattice
12.6 The Monte Carlo transfer matrix method
Exercises
References
13. The finite element method for partial differential equations
13.1 Introduction
13.2 The Poisson equation
13.3 Linear elasticity
13.4 Error estimators
13.5 Local refinement
13.6 Dynamical finite element method
13.7 Concurrent coupling of length scales: FEM and MD
Exercises
References
14. The lattice Boltzmann method for fluid dynamics
14.1 Introduction
14.2 Derivation of the Navier–Stokes equations
14.3 The lattice Boltzmann model
14.4 Additional remarks
14.5 Derivation of the Navier–Stokes equation from the lattice Boltzmann model
Exercises
References
15. Computational methods for lattice field theories
15.1 Introduction
15.2 Quantum field theory
15.3 Interacting fields and renormalisation
15.4 Algorithms for lattice field theories
15.5 Reducing critical slowing down
15.6 Comparison of algorithms for scalar field theory
15.7 Gauge field theories
Exercises
References
16. High performance computing and parallelism
16.1 Introduction
16.2 Pipelining
16.3 Parallelism
16.4 Parallel algorithms for molecular dynamics
References
Appendix A Numerical methods
A1 About numerical methods
A2 Iterative procedures for special functions
A3 Finding the root of a function
A4 Finding the optimum of a function
A5 Discretisation
A6 Numerical quadratures
A7 Differential equations
A8 Linear algebra problems
A9 The fast Fourier transform
Exercises
References
Appendix B Random number generators
B1 Random numbers and pseudo-random numbers
B2 Random number generators and properties of pseudo-random numbers
B3 Nonuniform random number generators
Exercises
References
Index
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