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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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