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A Guide to Monte Carlo Simulations in Statistical Physics

Book Description

Dealing with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics and statistical mechanics, this book provides an introduction to computer simulations in physics. This edition now contains material describing powerful new algorithms that have appeared since the previous edition was published, and highlights recent technical advances and key applications that these algorithms now make possible. Updates also include several new sections and a chapter on the use of Monte Carlo simulations of biological molecules. Throughout the book there are many applications, examples, recipes, case studies, and exercises to help the reader understand the material. It is ideal for graduate students and researchers, both in academia and industry, who want to learn techniques that have become a third tool of physical science, complementing experiment and analytical theory.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Introduction
    1. 1.1 What is a Monte Carlo simulation?
    2. 1.2 What problems can we solve with it?
    3. 1.3 What difficulties will we encounter?
      1. 1.3.1 Limited computer time and memory
      2. 1.3.2 Statistical and other errors
    4. 1.4 What strategy should we follow in approaching a problem?
    5. 1.5 How do simulations relate to theory and experiment?
    6. 1.6 Perspective
    7. References
  8. 2. Some necessary background
    1. 2.1 Thermodynamics and statistical mechanics: a quick reminder
      1. 2.1.1 Basic notions
      2. 2.1.2 Phase transitions
      3. 2.1.3 Ergodicity and broken symmetry
      4. 2.1.4 Fluctuations and the Ginzburg criterion
      5. 2.1.5 A standard exercise: the ferromagnetic Ising model
    2. 2.2 Probability theory
      1. 2.2.1 Basic notions
      2. 2.2.2 Special probability distributions and the central limit theorem
      3. 2.2.3 Statistical errors
      4. 2.2.4 Markov chains and master equations
      5. 2.2.5 The ‘art’ of random number generation
    3. 2.3 Non-equilibrium and dynamics: some introductory comments
      1. 2.3.1 Physical applications of master equations
      2. 2.3.2 Conservation laws and their consequences
      3. 2.3.3 Critical slowing down at phase transitions
      4. 2.3.4 Transport coefficients
      5. 2.3.5 Concluding comments
    4. References
  9. 3. Simple sampling Monte Carlo methods
    1. 3.1 Introduction
    2. 3.2 Comparisons of methods for numerical integration of given functions
      1. 3.2.1 Simple methods
      2. 3.2.2 Intelligent methods
    3. 3.3 Boundary value problems
    4. 3.4 Simulation of radioactive decay
    5. 3.5 Simulation of transport properties
      1. 3.5.1 Neutron transport
      2. 3.5.2 Fluid flow
    6. 3.6 The percolation problem
      1. 3.6.1 Site percolation
      2. 3.6.2 Cluster counting: the Hoshen–Kopelman algorithm
      3. 3.6.3 Other percolation models
    7. 3.7 Finding the groundstate of a Hamiltonian
    8. 3.8 Generation of ‘random’ walks
      1. 3.8.1 Introduction
      2. 3.8.2 Random walks
      3. 3.8.3 Self-avoiding walks
      4. 3.8.4 Growing walks and other models
    9. 3.9 Final remarks
    10. References
  10. 4. Importance sampling Monte Carlo methods
    1. 4.1 Introduction
    2. 4.2 The simplest case: single spin-flip sampling for the simple Ising model
      1. 4.2.1 Algorithm
      2. 4.2.2 Boundary conditions
      3. 4.2.3 Finite size effects
      4. 4.2.4 Finite sampling time effects
      5. 4.2.5 Critical relaxation
    3. 4.3 Other discrete variable models
      1. 4.3.1 Ising models with competing interactions
      2. 4.3.2 q-state Potts models
      3. 4.3.3 Baxter and Baxter–Wu models
      4. 4.3.4 Clock models
      5. 4.3.5 Ising spin glass models
      6. 4.3.6 Complex fluid models
    4. 4.4 Spin-exchange sampling
      1. 4.4.1 Constant magnetization simulations
      2. 4.4.2 Phase separation
      3. 4.4.3 Diffusion
      4. 4.4.4 Hydrodynamic slowing down
    5. 4.5 Microcanonical methods
      1. 4.5.1 Demon algorithm
      2. 4.5.2 Dynamic ensemble
      3. 4.5.3 Q2R
    6. 4.6 General remarks, choice of ensemble
    7. 4.7 Statics and dynamics of polymer models on lattices
      1. 4.7.1 Background
      2. 4.7.2 Fixed bond length methods
      3. 4.7.3 Bond fluctuation method
      4. 4.7.4 Enhanced sampling using a fourth dimension
      5. 4.7.5 The ‘wormhole algorithm’ – another method to equilibrate dense polymeric systems
      6. 4.7.6 Polymers in solutions of variable quality: θ-point, collapse transition, unmixing
      7. 4.7.7 Equilibrium polymers: a case study
    8. 4.8 Some advice
    9. References
  11. 5. More on importance sampling Monte Carlo methods for lattice systems
    1. 5.1 Cluster flipping methods
      1. 5.1.1 Fortuin–Kasteleyn theorem
      2. 5.1.2 Swendsen–Wang method
      3. 5.1.3 Wolff method
      4. 5.1.4 ‘Improved estimators’
      5. 5.1.5 Invaded cluster algorithm
      6. 5.1.6 Probability changing cluster algorithm
    2. 5.2 Specialized computational techniques
      1. 5.2.1 Expanded ensemble methods
      2. 5.2.2 Multispin coding
      3. 5.2.3 N-fold way and extensions
      4. 5.2.4 Hybrid algorithms
      5. 5.2.5 Multigrid algorithms
      6. 5.2.6 Monte Carlo on vector computers
      7. 5.2.7 Monte Carlo on parallel computers
    3. 5.3 Classical spin models
      1. 5.3.1 Introduction
      2. 5.3.2 Simple spin-flip method
      3. 5.3.3 Heatbath method
      4. 5.3.4 Low temperature techniques
      5. 5.3.5 Over-relaxation methods
      6. 5.3.6 Wolff embedding trick and cluster flipping
      7. 5.3.7 Hybrid methods
      8. 5.3.8 Monte Carlo dynamics vs. equation of motion dynamics
      9. 5.3.9 Topological excitations and solitons
    4. 5.4 Systems with quenched randomness
      1. 5.4.1 General comments: averaging in random systems
      2. 5.4.2 Parallel tempering: a general method to better equilibrate systems with complex energy landscapes
      3. 5.4.3 Random fields and random bonds
      4. 5.4.4 Spin glasses and optimization by simulated annealing
      5. 5.4.5 Ageing in spin glasses and related systems
      6. 5.4.6 Vector spin glasses: developments and surprises
    5. 5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study
    6. 5.6 Sampling the free energy and entropy
      1. 5.6.1 Thermodynamic integration
      2. 5.6.2 Groundstate free energy determination
      3. 5.6.3 Estimation of intensive variables: the chemical potential
      4. 5.6.4 Lee–Kosterlitz method
      5. 5.6.5 Free energy from finite size dependence at Tc
    7. 5.7 Miscellaneous topics
      1. 5.7.1 Inhomogeneous systems: surfaces, interfaces, etc.
      2. 5.7.2 Other Monte Carlo schemes
      3. 5.7.3 Inverse and reverse Monte Carlo methods
      4. 5.7.4 Finite size effects: a review and summary
      5. 5.7.5 More about error estimation
      6. 5.7.6 Random number generators revisited
    8. 5.8 Summary and perspective
    9. References
  12. 6. Off-lattice models
    1. 6.1 Fluids
      1. 6.1.1 NVT ensemble and the virial theorem
      2. 6.1.2 NpT ensemble
      3. 6.1.3 Grand canonical ensemble
      4. 6.1.4 Near critical coexistence: a case study
      5. 6.1.5 Subsystems: a case study
      6. 6.1.6 Gibbs ensemble
      7. 6.1.7 Widom particle insertion method and variants
      8. 6.1.8 Monte Carlo Phase Switch
      9. 6.1.9 Cluster algorithm for fluids
    2. 6.2 ‘Short range’ interactions
      1. 6.2.1 Cutoffs
      2. 6.2.2 Verlet tables and cell structure
      3. 6.2.3 Minimum image convention
      4. 6.2.4 Mixed degrees of freedom reconsidered
    3. 6.3 Treatment of long range forces
      1. 6.3.1 Reaction field method
      2. 6.3.2 Ewald method
      3. 6.3.3 Fast multipole method
    4. 6.4 Adsorbed monolayers
      1. 6.4.1 Smooth substrates
      2. 6.4.2 Periodic substrate potentials
    5. 6.5 Complex fluids
      1. 6.5.1 Application of the Liu–Luijten algorithm to a binary fluid mixture
    6. 6.6 Polymers: an introduction
      1. 6.6.1 Length scales and models
      2. 6.6.2 Asymmetric polymer mixtures: a case study
      3. 6.6.3 Applications: dynamics of polymer melts; thin adsorbed polymeric films
      4. 6.6.4 Polymer melts: speeding up bond fluctuation model simulations
    7. 6.7 Configurational bias and ‘smart Monte Carlo’
    8. 6.8 Outlook
    9. References
  13. 7. Reweighting methods
    1. 7.1 Background
      1. 7.1.1 Distribution functions
      2. 7.1.2 Umbrella sampling
    2. 7.2 Single histogram method: the Ising model as a case study
    3. 7.3 Multihistogram method
    4. 7.4 Broad histogram method
    5. 7.5 Transition matrix Monte Carlo
    6. 7.6 Multicanonical sampling
      1. 7.6.1 The multicanonical approach and its relationship to canonical sampling
      2. 7.6.2 Near first order transitions
      3. 7.6.3 Groundstates in complicated energy landscapes
      4. 7.6.4 Interface free energy estimation
    7. 7.7 A case study: the Casimir effect in critical systems
    8. 7.8 ‘Wang–Landau sampling’
      1. 7.8.1 Basic algorithm
      2. 7.8.2 Applications to models with continuous variables
      3. 7.8.3 Case studies with two-dimensional Wang–Landau sampling
      4. 7.8.4 Back to numerical integration
    9. 7.9 A case study: evaporation/condensation transition of droplets
    10. References
  14. 8. Quantum Monte Carlo methods
    1. 8.1 Introduction
    2. 8.2 Feynman path integral formulation
      1. 8.2.1 Off-lattice problems: low-temperature properties of crystals
      2. 8.2.2 Bose statistics and superfluidity
      3. 8.2.3 Path integral formulation for rotational degrees of freedom
    3. 8.3 Lattice problems
      1. 8.3.1 The Ising model in a transverse field
      2. 8.3.2 Anisotropic Heisenberg chain
      3. 8.3.3 Fermions on a lattice
      4. 8.3.4 An intermezzo: the minus sign problem
      5. 8.3.5 Spinless fermions revisited
      6. 8.3.6 Cluster methods for quantum lattice models
      7. 8.3.7 Continuous time simulations
      8. 8.3.8 Decoupled cell method
      9. 8.3.9 Handscomb’s method
      10. 8.3.10 Wang–Landau sampling for quantum models
      11. 8.3.11 Fermion determinants
    4. 8.4 Monte Carlo methods for the study of groundstate properties
      1. 8.4.1 Variational Monte Carlo (VMC)
      2. 8.4.2 Green’s function Monte Carlo methods (GFMC)
    5. 8.5 Concluding remarks
    6. References
  15. 9. Monte Carlo renormalization group methods
    1. 9.1 Introduction to renormalization group theory
    2. 9.2 Real space renormalization group
    3. 9.3 Monte Carlo renormalization group
      1. 9.3.1 Large cell renormalization
      2. 9.3.2 Ma’s method: finding critical exponents and the fixed point Hamiltonian
      3. 9.3.3 Swendsen’s method
      4. 9.3.4 Location of phase boundaries
      5. 9.3.5 Dynamic problems: matching time-dependent correlation functions
      6. 9.3.6 Inverse Monte Carlo renormalization group transformations
    4. References
  16. 10. Non-equilibrium and irreversible processes
    1. 10.1 Introduction and perspective
    2. 10.2 Driven diffusive systems (driven lattice gases)
    3. 10.3 Crystal growth
    4. 10.4 Domain growth
    5. 10.5 Polymer growth
      1. 10.5.1 Linear polymers
      2. 10.5.2 Gelation
    6. 10.6 Growth of structures and patterns
      1. 10.6.1 Eden model of cluster growth
      2. 10.6.2 Diffusion limited aggregation
      3. 10.6.3 Cluster–cluster aggregation
      4. 10.6.4 Cellular automata
    7. 10.7 Models for film growth
      1. 10.7.1 Background
      2. 10.7.2 Ballistic deposition
      3. 10.7.3 Sedimentation
      4. 10.7.4 Kinetic Monte Carlo and MBE growth
    8. 10.8 Transition path sampling
    9. 10.9 Forced polymer pore translocation: a case study
    10. 10.10 Outlook: variations on a theme
    11. References
  17. 11. Lattice gauge models: a brief introduction
    1. 11.1 Introduction: gauge invariance and lattice gauge theory
    2. 11.2 Some technical matters
    3. 11.3 Results for Z(N) lattice gauge models
    4. 11.4 Compact U(1) gauge theory
    5. 11.5 SU(2) lattice gauge theory
    6. 11.6 Introduction: quantum chromodynamics (QCD) and phase transitions of nuclear matter
    7. 11.7 The deconfinement transition of QCD
    8. 11.8 Towards quantitative predictions
    9. References
  18. 12. A brief review of other methods of computer simulation
    1. 12.1 Introduction
    2. 12.2 Molecular dynamics
      1. 12.2.1 Integration methods (microcanonical ensemble)
      2. 12.2.2 Other ensembles (constant temperature, constant pressure, etc.)
      3. 12.2.3 Non-equilibrium molecular dynamics
      4. 12.2.4 Hybrid methods (MD + MC)
      5. 12.2.5 Ab initio molecular dynamics
      6. 12.2.6 Hyperdynamics and metadynamics
    3. 12.3 Quasi-classical spin dynamics
    4. 12.4 Langevin equations and variations (cell dynamics)
    5. 12.5 Micromagnetics
    6. 12.6 Dissipative particle dynamics (DPD)
    7. 12.7 Lattice gas cellular automata
    8. 12.8 Lattice Boltzmann equation
    9. 12.9 Multiscale simulation
    10. References
  19. 13. Monte Carlo simulations at the periphery of physics and beyond
    1. 13.1 Commentary
    2. 13.2 Astrophysics
    3. 13.3 Materials science
    4. 13.4 Chemistry
    5. 13.5 ‘Biologically inspired’ physics
      1. 13.5.1 Commentary and perspective
      2. 13.5.2 Lattice proteins
      3. 13.5.3 Cell sorting
    6. 13.6 Biology
    7. 13.7 Mathematics/statistics
    8. 13.8 Sociophysics
    9. 13.9 Econophysics
    10. 13.10 ‘Traffic’ simulations
    11. 13.11 Medicine
    12. 13.12 Networks: what connections really matter?
    13. 13.13 Finance
    14. References
  20. 14. Monte Carlo studies of biological molecules
    1. 14.1 Introduction
    2. 14.2 Protein folding
      1. 14.2.1 Introduction
      2. 14.2.2 How to best simulate proteins: Monte Carlo or molecular dynamics
      3. 14.2.3 Generalized ensemble methods
      4. 14.2.4 Globular proteins: a case study
      5. 14.2.5 Simulations of membrane proteins
    3. 14.3 Monte Carlo simulations of carbohydrates
    4. 14.4 Determining macromolecular structures
    5. 14.5 Outlook
    6. References
  21. 15. Outlook
  22. Appendix: listing of programs mentioned in the text
  23. Index