Computational Physics

by Jos Thijssen

• ISBN13: 9780521833462
• ISBN10: 0521833469
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-04-09
• Publisher: Cambridge University Press

Summary

This Second Edition has been fully updated. The wide range of topics covered in the First Edition has been extended with new chapters on finite element methods and lattice Boltzmann simulation. New sections have been added to the chapters on density functional theory, quantum molecular dynamics, Monte Carlo simulation and diagonalisation of one-dimensional quantum systems.

Author Biography

Jos Thijssen is a lecturer at the Kavli Institute of Nanoscience at Delft University of Technology

Table of Contents

Preface to the first edition	p. xi
Preface to the second edition	p. xiv
Introduction	p. 1
Physics and computational physics	p. 1
Classical mechanics and statistical mechanics	p. 1
Stochastic simulations	p. 4
Electrodynamics and hydrodynamics	p. 5
Quantum mechanics	p. 6
Relations between quantum mechanics and classical statistical physics	p. 7
Quantum molecular dynamics	p. 8
Quantum field theory	p. 9
About this book	p. 9
Exercises	p. 11
References	p. 13
Quantum scattering with a spherically symmetric potential	p. 14
Introduction	p. 14
A program for calculating cross sections	p. 18
Calculation of scattering cross sections	p. 25
Exercises	p. 27
References	p. 28
The variational method for the Schrodinger equation	p. 29
Variational calculus	p. 29
Examples of variational calculations	p. 32
Solution of the generalised eigenvalue problem	p. 36
Perturbation theory and variational calculus	p. 37
Exercises	p. 39
References	p. 41
The Hartree-Fock method	p. 43
Introduction	p. 43
The Born-Oppenheimer approximation and the independent-particle method	p. 44
The helium atom	p. 46
Many-electron systems and the Slater determinant	p. 52
Self-consistency and exchange: Hartree-Fock theory	p. 54
Basis functions	p. 60
The structure of a Hartree-Fock computer program	p. 69
Integrals involving Gaussian functions	p. 73
Applications and results	p. 77
Improving upon the Hartree-Fock approximation	p. 78
Exercises	p. 80
References	p. 87
Density functional theory	p. 89
Introduction	p. 89
The local density approximation	p. 95
Exchange and correlation: a closer look	p. 97
Beyond DFT: one-and two-particle excitations	p. 101
A density functional program for the helium atom	p. 109
Applications and results	p. 114
Exercises	p. 116
References	p. 119
Solving the Schrodinger equation in periodic solids	p. 122
Introduction: definitions	p. 123
Band structures and Bloch's theorem	p. 124
Approximations	p. 126
Band structure methods and basis functions	p. 133
Augmented plane wave methods	p. 135
The linearised APW (LAPW) method	p. 141
The pseudopotential method	p. 144
Extracting information from band structures	p. 160
Some additional remarks	p. 162
Other band methods	p. 163
Exercises	p. 163
References	p. 167
Classical equilibrium statistical mechanics	p. 169
Basic theory	p. 169
Examples of statistical models; phase transitions	p. 176
Phase transitions	p. 184
Determination of averages in simulations	p. 192
Exercises	p. 194
References	p. 195
Molecular dynamics simulations	p. 197
Introduction	p. 197
Molecular dynamics at constant energy	p. 200
A molecular dynamics simulation program for argon	p. 208
Integration methods: symplectic integrators	p. 211
Molecular dynamics methods for different ensembles	p. 223
Molecular systems	p. 232
Long-range interactions	p. 241
Langevin dynamics simulation	p. 247
Dynamical quantities: nonequilibrium molecular dynamics	p. 251
Exercises	p. 253
References	p. 259
Quantum molecular dynamics	p. 263
Introduction	p. 263
The molecular dynamics method	p. 266
An example: quantum molecular dynamics for the hydrogen molecule	p. 272
Orthonormalisation; conjugate gradient and RM-DIIS techniques	p. 278
Implementation of the Car-Parrinello technique for pseudopotential DFT	p. 289
Exercises	p. 290
References	p. 293
The Monte Carlo method	p. 295
Introduction	p. 295
Monte Carlo integration	p. 296
Importance sampling through Markov chains	p. 299
Other ensembles	p. 310
Estimation of free energy and chemical potential	p. 316
Further application and Monte Carlo methods	p. 319
The temperature of a finite system	p. 330
Exercises	p. 334
References	p. 335
Transfer matrix and diagonalisation of spin chains	p. 338
Introduction	p. 338
The one-dimensional Ising model and the transfer matrix	p. 339
Two-dimensional spin models	p. 343
More complicated models	p. 347
'Exact' diagonalisation of quantum chains	p. 349
Quantum renormalisation in real space	p. 355
The density matrix renormalisation group method	p. 358
Exercises	p. 365
References	p. 370
Quantum Monte Carlo methods	p. 372
Introduction	p. 372
The variational Monte Carlo method	p. 373
Diffusion Monte Carlo	p. 387
Path-integral Monte Carlo	p. 398
Quantum Monte Carlo on a lattice	p. 410
The Monte Carlo transfer matrix method	p. 414
Exercises	p. 417
References	p. 421
The finite element method for partial differential equations	p. 423
Introduction	p. 423
The Poisson equation	p. 424
Linear elasticity	p. 429
Error estimators	p. 434
Local refinement	p. 436
Dynamical finite element method	p. 439
Concurrent coupling of length scales: FEM and MD	p. 440
Exercises	p. 445
References	p. 446
The lattice Boltzmann method for fluid dynamics	p. 448
Introduction	p. 448
Derivation of the Navier-Stokes equations	p. 449
The lattice Boltzmann model	p. 455
Additional remarks	p. 458
Derivation of the Navier-Stokes equation from the lattice Boltzmann model	p. 460
Exercises	p. 463
References	p. 454
Computational methods for lattice field theories	p. 466
Introduction	p. 466
Quantum field theory	p. 467
Interacting fields and renormalisation	p. 473
Algorithms for lattice field theories	p. 477
Reducing critical slowing down	p. 491
Comparison of algorithms for scalar field theory	p. 509
Gauge field theories	p. 510
Exercises	p. 532
References	p. 536
High performance computing and parallelism	p. 540
Introduction	p. 540
Pipelining	p. 541
Parallelism	p. 545
Parallel algorithms for molecular dynamics	p. 552
References	p. 556
Numerical methods	p. 557
About numerical methods	p. 557
Iterative procedures for special functions	p. 558
Finding the root of a function	p. 559
Finding the optimum of a function	p. 560
Discretisation	p. 565
Numerical quadratures	p. 566
Differential equations	p. 568
Linear algebra problems	p. 590
The fast Fourier transform	p. 598
Exercises	p. 601
References	p. 603
Random number generators	p. 605
Random numbers and pseudo-random numbers	p. 605
Random number generators and properties of pseudo-random numbers	p. 606
Nonuniform random number generators	p. 609
Exercises	p. 611
References	p. 612
Index	p. 613
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