Chapter
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: why bother about dynamics when doing Monte Carlo for statics?
3 Simple sampling Monte Carlo methods
3.2 Comparisons of methods for numerical integration of given functions
3.2.2 Intelligent methods
3.3 Boundary value problems
3.4 Simulation of radioactive decay
3.5 Simulation of transport properties
3.6 The percolation problem
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
3.8.3 Self-avoiding walks
3.8.4 Growing walks and other models
4 Importance sampling Monte Carlo methods
4.2 The simplest case: single spin-flip sampling for the simple Ising model
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
4.3.5 Ising spin glass models
4.3.6 Complex fluid models
4.4 Spin-exchange sampling
4.4.1 Constant magnetization simulations
4.4.4 Hydrodynamic slowing down
4.5 Microcanonical methods
4.6 General remarks, choice of ensemble
4.7 Statics and dynamics of polymer models on lattices
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: -point, collapse transition, unmixing
4.7.7 Equilibrium polymers: a case study
4.7.8 The pruned enriched Rosenbluth method (PERM): a biased sampling approach to simulate very long isolated chains
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
5.1.4 ‘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
5.2.3 N-fold way and extensions
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
5.3.2 Simple spin-flip method
5.3.4 Low temperature techniques
5.3.5 Over-relaxation methods
5.3.6 Wolff embedding trick and cluster flipping
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 Methods for systems with long range interactions
5.7 Parallel tempering, simulated tempering, and related methods: accuracy considerations
5.8 Sampling the free energy and entropy
5.8.1 Thermodynamic integration
5.8.2 Groundstate free energy determination
5.8.3 Estimation of intensive variables: the chemical potential
5.8.4 Lee-Kosterlitz method
5.8.5 Free energy from finite size dependence at Tc
5.9.1 Inhomogeneous systems: surfaces, interfaces, etc.
5.9.2 Anisotropic critical phenomena: simulation boxes with arbitrary aspect ratio
5.9.3 Other Monte Carlo schemes
5.9.4 Inverse and reverse Monte Carlo methods
5.9.5 Finite size effects: review and summary
5.9.6 More about error estimation
5.9.7 Random number generators revisited
5.10 Summary and perspective
6.1.1 NVT ensemble and the virial theorem
6.1.3 Grand canonical ensemble
6.1.4 Near critical coexistence: a case study
6.1.5 Subsystems: a case study
6.1.7 Widom particle insertion method and variants
6.1.8 Monte Carlo phase switch
6.1.9 Cluster algorithm for fluids
6.1.10 Event chain algorithms
6.2 ‘Short range’ interactions
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
6.3.3 Fast multipole method
6.4.2 Periodic substrate potentials
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
6.6.4 Polymer melts: speeding up bond fluctuation model simulations
6.7 Configurational bias and ‘smart Monte Carlo’
6.8 Estimation of excess free energies due to walls for fluids and solids
6.9 A symmetric, Lennard-Jones mixture: a case study
6.10 Finite size effects on interfacial properties: a case study
7.1.1 Distribution functions
7.2 Single histogram method
7.2.1 The Ising model as a case study
7.2.2 The surface-bulk multicritical point: another case study
7.3 Multihistogram 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
7.8.2 Applications to models with continuous variables
7.8.3 A simple example of two-dimensional Wang-Landau sampling
7.8.4 Microcanonical entropy inflection points
7.8.5 Back to numerical integration
7.8.6 Replica exchange Wang-Landau sampling
7.9 A case study: evaporation/condensation transition of droplets
8 Quantum Monte Carlo methods
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
8.3.1 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 and the stochastic series expansion (SSE) approach
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 Towards constructing the nodal surface of off-lattice, many- fermion systems: the ‘survival of the fittest’ algorithm
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
9.3.4 Location of phase boundaries
9.3.5 Dynamic problems: matching time-dependent correlation functions
9.3.6 Inverse Monte Carlo renormalization group transformations
10 Non-equilibrium and irreversible processes
10.1 Introduction and perspective
10.2 Driven diffusive systems (driven lattice gases)
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
10.7 Models for film growth
10.7.2 Ballistic deposition
10.7.4 Kinetic Monte Carlo and MBE growth
10.8 Transition path sampling
10.9 Forced polymer pore translocation: a case study
10.10 The Jarzynski non-equilibrium work theorem and its application to obtain free energy differences from trajectories
10.11 Outlook: variations on a theme
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 Towards quantitative predictions
11.9 Density of states in gauge theories
12 A brief review of other methods of computer simulation
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
12.2.6 Hyperdynamics and metadynamics
12.3 Quasi-classical spin dynamics
12.4 Langevin equations and variations (cell dynamics)
12.6 Dissipative particle dynamics (DPD)
12.7 Lattice gas cellular automata
12.8 Lattice Boltzmann equation
12.9 Multiscale simulation
12.10 Multiparticle collision dynamics
13 Monte Carlo simulations at the periphery of physics and beyond
13.5 ‘Biologically inspired’ physics
13.5.1 Commentary and perspective
13.7 Mathematics/statistics
13.10 ‘Traffic’ simulations
13.12 Networks: what connections really matter?
14 Monte Carlo studies of biological molecules
14.2.2 How to best simulate proteins: Monte Carlo or molecular dynamics?
14.2.3 Generalized ensemble methods
14.2.4 Globular proteins: a case study
14.2.5 Simulations of membrane proteins
14.3 Monte Carlo simulations of RNA structures
14.4 Monte Carlo simulations of carbohydrates
14.5 Determining macromolecular structures
Appendix: Listing of programs mentioned in the text
A Guide to Monte Carlo Simulations in Statistical Physics
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