Preface
1 Introduction
1.1 What is a Monte Carlo simulation？
1.2 What problems can we solve with it？
1.3 What difficulties will we encounter？
1.3.1 Limited computer time and memory
1.3.2 Statistical and other errors
1.4 What strategy should we follow in approaching a problem？
1.5 How do simulations relate to theory and experiment？
1.6 Perspective
2 Some necessary background
2.1 Thermodynamics and statistical mechanics: a quick reminder
2.1.1 Basic notions
2.1.2 Phase transitions
2.1.3 Ergodicity and broken symmetry
2.1.4 Fluctuations and the Ginzburg criterion
2.1.5 A standard exercise: the ferromagnetic Ising model
2.2 Probability theory
2.2.1 Basic notions
2.2.2 Special probability distributions and the central limit theorem
2.2.3 Statistical errors
2.2.4 Markov chains and master equations
2.2.5 The 'art' of random number generation
2.3 Non-equilibrium and dynamics: some introductory comments
2.3.1 Physical applications of master equations
2.3.2 Conservation laws and their consequences
2.3.3 Critical slowing down at phase transitions
2.3.4 Transport coefficients
2.3.5 Concluding comments
References
3 Simple sampling Monte Carlo methods
3.1 Introduction
3.2 Comparisons of methods for numerical integration of given
functions
3.2.1 Simple methods
3.2.2 Intelligent methods
3.3 Boundary value problems
3.4 Simulation of radioactive decay
3.5 Simulation of transp6rt properties
3.5.1 Neutron transport
3.5.2 Fluid flow
3.6 The percolation problem
3.6.1 Site percolation
3.6.2 Cluster counting: the Hoshen-Kopelman algorithm
3.6.3 Other percolation models
3.7 Finding the groundstate of a Hamiltonian
3.8 Generation of 'random' walks
3.8.1 Introduction
3.8.2 Random walks
3.8.3 Self-avoiding walks
3.8.4 Growing walks and other models
3.9 Final remarks
References
4 Importance sampling Monte Carlo methods
4.1 Introduction
4.2 The simplest case: single spin-flip sampling for the simple Ising model
4.2.1 Algorithm
4.2.2 Boundary conditions
4.2.3 Finite size effects
4.2.4 Finite sampling time effects
4.2.5 Critical relaxation
4.3 Other discrete variable models
4.3.1 Ising models with competing interactions
4.3.2 q-state Potts models
4.3.3 Baxter and Baxter-Wu models
4.3.4 Clock models
4.3.5 Ising spin glass models
4.3.6 Complex fluid models
4.4 Spin-exchange sampling
4.4.1 Constant magnetization simulations
4.4.2 Phase separation
4.4.3 Diffusion
4.4.4 Hydrodynamic slowing down
4.5 Microcanonical methods
4.5.1 Demon algorithm
4.5.2 Dynamic ensemble
4.5.3 Q2R
4.6 General remarks, choice of ensemble
4.7 Statics and dynamics of polymer models on lattices
4.7.1 Background
4.7.2 Fixed bond length methods
4.7.3 Bond fluctuation method
4.7.4 Enhanced sampling using a fourth dimension
4.7.5 The 'wormhole algorithm' - another method to equilibrate dense polymeric systems
4.7.6 Polymers in solutions of variable quality: 0-point, collapse transition, unmixing
4.7.7 Equilibrium polymers: a case study
4.8 Some advice
References
5 More on importance sampling Monte Carlo methods for lattice systems
5.1 Cluster flipping methods
5.1.1 Fortuin-Kasteleyn theorem
5.1.2 Swendsen-Wang method
5.1.3 Wolff method
5.1.4 'Improved estimators'
5.1.5 Invaded cluster algorithm
5.1.6 Probability changing cluster algorithm
5.2 Specialized computational techniques
5.2.1 Expanded ensemble methods
5.2.2 Multispin coding
5.2.3 N-fold way and extensions
5.2.4 Hybrid algorithms
5.2.5 Multigrid algorithms
5.2.6 Monte Carlo on vector computers
5.2.7 Monte Carlo on parallel computers
5.3 Classical spin models
5.3.1 Introduction
5.3.2 Simple spin-flip method
5.3.3 Heatbath method
5.3.4 Low temperature techniques
5.3.5 Over-relaxation methods
5.3.6 Wolff embedding trick and cluster flipping
5.3.7 Hybrid methods
5.3.8 Monte Carlo dynamics vs. equation of motion dynamics
5.3.9 Topological excitations and solitons
5.4 Systems with quenched randomness
5.4.1 General comments: averaging in random systems
5.4.2 Parallel tempering: a general method to better equilibrate systems with complex energy landscapes
5.4.3 Random fields and random bonds
5.4.4 Spin glasses and optimization by simulated annealing
5.4.5 Ageing in spin glasses and related systems
5.4.6 Vector spin glasses: developments and surprises
5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study
5.6 Sampling the free energy and entropy
5.6.1 Thermodynamic integration
5.6.2 Groundstate free energy determination
5.6.3 Estimation of intensive variables: the chemical potential
5.6.4 Lee-Kosterlitz method
5.6.5 Free energy from finite size dependence at Tc
5.7 Miscellaneous topics
5.7.1 Inhomogeneous systems: surfaces, interfaces, etc.
5.7.2 Other Monte Carlo schemes
5.7.3 Inverse Monte Carlo methods
5.7.4 Finite size effects: a review and summary
5.7.5 More about error estimation
5.7.6 Random number generators revisited
5.8 Summary and perspective
References
6 Off-lattice models
6.1 Fluids
6.1.1 NVT ensemble and the virial theorem
6.1.2 NpT ensemble
6.1.3 Grand canonical ensemble
6.1.4 Near critical coexistence: a case study
6.1.5 Subsystems: a case study
6.1.6 Gibbs ensemble
6.1.7 Widom particle insertion method and variants
6.1.8 Monte Carlo Phase Switch
6.1.9 Cluster algorithm for fluids
6.2 'Short range' interactions
6.2.1 Cutoffs
6.2.2 Verlet tables and cell structure
6.2.3 Minimum image convention
6.2.4 Mixed degrees of freedom reconsidered
6.3 Treatment of long range forces
6.3.1 Reaction field method
6.3.2 Ewald method
6.3.3 Fast multipole method
6.4 Adsorbed monolayers
6.4.1 Smooth substrates
6.4.2 Periodic substrate potentials
6.5 Complex fluids
6.5.1 Application of the Liu-Luijten algorithm to a binary fluid mixture
6.6 Polymers: an introduction
6.6.1 Length scales and models
6.6.2 Asymmetric polymer mixtures: a case study
6.6.3 Applications: dynamics of polymer melts; thin adsorbed polymeric films
6.7 Configurational bias and 'smart Monte Carlo'
References
7 Reweighting methods
7.1 Background
7.1.1 Distribution functions
7.1.2 Umbrella sampling
7.2 Single histogram method: the Ising model as a case study
7.3 Multi-histogram method
7.4 Broad histogram method
7.5 Transition matrix Monte Carlo
7.6 Multicanonical sampling
7.6.1 The multicanonical approach and its relationship to canonical sampling
7.6.2 Near first order transitions
7.6.3 Groundstates in complicated energy landscapes
7.6.4 Interface free energy estimation
7.7 A case study: the Casimir effect in critical systems
7.8 'Wang-Landau sampling'
7.9 A case study: evaporation/condensation transition of droplets
References
8 Quantum Monte Carlo methods
8.1 Introduction
8.2 Feynman path integral formulation
8.2.1 Off-lattice problems: low-temperature properties of crystals
8.2.2 Bose statistics and superfluidity
8.2.3 Path integral formulation for rotational degrees of freedom
8.3 Lattice problems
8.3.1 The Ising model in a transverse field
8.3.2 Anisotropic Heisenberg chain
8.3.3 Fermions on a lattice
8.3.4 An intermezzo: the minus sign problem
8.3.5 Spinless fermions revisited
8.3.6 Cluster methods for quantum lattice models
8.3.7 Continuous time simulations
8.3.8 Decoupled cell method
8.3.9 Handscomb's method
8.3.10 Wang-Landau sampling for quantum models
8.3.11 Fermion determinants
8.4 Monte Carlo methods for the study of groundstate properties
8.4.1 Variational Monte Carlo （VMC）
8.4.2 Green's function Monte Carlo methods （GFMC）
8.5 Concluding remarks
References
9 Monte Carlo renormalization group methods
9.1 Introduction to renormalization group theory
9.2 Real space renormalization group
9.3 Monte Carlo renormalization group
9.3.1 Large cell renormalization
9.3.2 Ma's method: finding critical exponents and the fixed point Hamiltonian
9.3.3 Swendsen's method
9.3.4 Location of phase boundaries
9.3.5 Dynamic problems: matching time-dependent correlation functions
9.3.6 Inverse Monte Carlo renormalization group transformations
References
10 Non-equilibrium and irreversible processes
10.1 Introduction and perspective
10.2 Driven diffusive systems （driven lattice gases）
10.3 Crystal growth
10.4 Domain growth
10.5 Polymer growth
10.5.1 Linear polymers
10.5.2 Gelation
10.6 Growth of structures and patterns
10.6.1 Eden model of cluster growth
10.6.2 Diffusion limited aggregation
10.6.3 Cluster-cluster aggregation
10.6.4 Cellular automata
10.7 Models for film growth
10.7.1 Background
10.7.2 Ballistic deposition
10.7.3 Sedimentation
10.7.4 Kinetic Monte Carlo and MBE growth
10.8 Transition path sampling
10.9 Outlook: variations on a theme
References
11 Lattice gauge models: a brief introduction
11.1 Introduction: gauge invariance and lattice gauge theory
11.2 Some technical matters
11.3 Results for Z（N） lattice gauge models
11.4 Compact U（1） gauge theory
11.5 SU（2） lattice gauge theory
11.6 Introduction: quantum chromodynamics （QCD） and phase transitions of nuclear matter
11.7 The deconfinement transition of QCD
11.8 Where are we now？
References
12 A brief review of other methods of computer simulation
12.1 Introduction
12.2 Molecular dynamics
12.2.1 Integration methods （microcanonical ensemble）
12.2.2 Other ensembles （constant temperature, constant pressure,etc.）
12.2.3 Non-equilibrium molecular dynamics
12.2.4 Hybrid methods （MD + MC）
12.2.5 Ab initio molecular dynamics
12.3 Quasi-classical spin dynamics
12.4 Langevin equations and variations （cell dynamics）
12.5 Micromagnetics
12.6 Dissipative particle dynamics （DPPD）
12.7 Lattice gas cellular automata
12.8 Lattice Boltzmann Equation
12.9 Multiscale simulation
References
13 Monte Carlo methods outside of physics
13.1 Commentary
13.2 Protein folding
13.2.1 Introduction
13.2.2 Generalized ensemble methods
13.2.3 Globular proteins: a case study
13.3 'Biologically inspired physics'
13.4 Mathematics/statistics
13.5 Sociophysics
13.6 Econophysics
13.7 'Traffic' simulations
13.8 Medicine
References
14 Outlook
Appendix: listing of programs mentioned in the text
Index
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