A Guide to Monte Carlo Simulations in Statistical Physics

by David P. Landau , Kurt Binder

• ISBN13: 9780521842389
• ISBN10: 0521842387
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-09-19
• Publisher: Cambridge University Press

Summary

This new and updated edition deals with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics, statistical mechanics, and related fields. After briefly recalling essential background in statistical mechanics and probability theory, it gives a succinct overview of simple sampling methods. The concepts behind the simulation algorithms are explained comprehensively, as are the techniques for efficient evaluation of system configurations generated by simulation. It contains many applications, examples, and exercises to help the reader and provides many new references to more specialized literature. This edition includes a brief overview of other methods of computer simulation and an outlook for the use of Monte Carlo simulations in disciplines beyond physics. This is an excellent guide for graduate students and researchers who use computer simulations in their research. It can be used as a textbook for graduate courses on computer simulations in physics and related disciplines.

Table of Contents

Preface	xii
Introduction	1	(6)
What is a Monte Carlo simulation?	1	(1)
What problems can we solve with it?	2	(1)
What difficulties will we encounter?	3	(1)
Limited computer time and memory	3	(1)
Statistical and other errors	3	(1)
What strategy should we follow in approaching a problem?	4	(1)
How do simulations relate to theory and experiment?	4	(2)
Perspective	6	(1)
Some necessary background	7	(41)
Thermodynamics and statistical mechanics: a quick reminder	7	(20)
Basic notions	7	(6)
Phase transitions	13	(11)
Ergodicity and broken symmetry	24	(1)
Fluctuations and the Ginzburg criterion	25	(1)
A standard exercise: the ferromagnetic Ising model	25	(2)
Probability theory	27	(12)
Basic notions	27	(2)
Special probability distributions and the central limit theorem	29	(1)
Statistical errors	30	(1)
Markov chains and master equations	31	(1)
The `art' of random number generation	32	(7)
Non-equilibrium and dynamics: some introductory comments	39	(9)
Physical applications of master equations	39	(1)
Conservation laws and their consequences	40	(3)
Critical slowing down at phase transitions	43	(2)
Transport coefficients	45	(1)
Concluding comments	45	(1)
References	45	(3)
Simple sampling Monte Carlo methods	48	(20)
Introduction	48	(1)
Comparisons of methods for numerical integration of given functions	48	(3)
Simple methods	48	(2)
Intelligent methods	50	(1)
Boundary value problems	51	(2)
Simulation of radioactive decay	53	(1)
Simulation of transport properties	54	(2)
Neutron transport	54	(1)
Fluid flow	55	(1)
The percolation problem	56	(4)
Site percolation	56	(3)
Cluster counting: the Hoshen--Kopelman algorithm	59	(1)
Other percolation models	60	(1)
Finding the groundstate of a Hamiltonian	60	(1)
Generation of `random' walks	61	(5)
Introduction	61	(1)
Random walks	62	(1)
Self-avoiding walks	63	(2)
Growing walks and other models	65	(1)
Final remarks	66	(2)
References	66	(2)
Importance sampling Monte Carlo methods	68	(69)
Introduction	68	(1)
The simplest case: single spin-flip sampling for the simple Ising model	69	(36)
Algorithm	70	(4)
Boundary conditions	74	(3)
Finite size effects	77	(13)
Finite sampling time effects	90	(8)
Critical relaxation	98	(7)
Other discrete variable models	105	(10)
Ising models with competing interactions	105	(4)
q-state Potts models	109	(1)
Baxter and Baxter--Wu models	110	(1)
Clock models	111	(2)
Ising spin glass models	113	(1)
Complex fluid models	114	(1)
Spin-exchange sampling	115	(5)
Constant magnetization simulations	115	(1)
Phase separation	115	(2)
Diffusion	117	(3)
Hydrodynamic slowing down	120	(1)
Microcanonical methods	120	(2)
Demon algorithm	120	(1)
Dynamic ensemble	121	(1)
Q2R	121	(1)
General remarks, choice of ensemble	122	(1)
Statics and dynamics of polymer models on lattices	122	(11)
Background	122	(1)
Fixed bond length methods	123	(1)
Bond fluctuation method	124	(1)
Enhanced sampling using a fourth dimension	125	(2)
The `wormhole algorithm' -- another method to equilibrate dense polymeric systems	127	(1)
Polymers in solutions of variable quality: θ-point, collapse transition, unmixing	127	(3)
Equilibrium polymers: a case study	130	(3)
Some advice	133	(4)
References	134	(3)
More on importance sampling Monte Carlo methods for lattice systems	137	(57)
Cluster flipping methods	137	(7)
Fortuin--Kasteleyn theorem	137	(1)
Swendsen--Wang method	138	(3)
Wolff method	141	(1)
`Improved estimators'	142	(1)
Invaded cluster algorithm	142	(1)
Probability changing cluster algorithm	143	(1)
Specialized computational techniques	144	(6)
Expanded ensemble methods	144	(1)
Multispin coding	144	(1)
N-fold way and extensions	145	(3)
Hybrid algorithms	148	(1)
Multigrid algorithms	148	(1)
Monte Carlo on vector computers	148	(1)
Monte Carlo on parallel computers	149	(1)
Classical spin models	150	(10)
Introduction	150	(1)
Simple spin-flip method	151	(2)
Heatbath method	153	(1)
Low temperature techniques	153	(1)
Over-relaxation methods	154	(1)
Wolff embedding trick and cluster flipping	154	(1)
Hybrid methods	155	(1)
Monte Carlo dynamics vs. equation of motion dynamics	156	(1)
Topological excitations and solitons	156	(4)
Systems with quenched randomness	160	(11)
General comments: averaging in random systems	160	(3)
Parallel tempering: a general method to better equilibrate systems with complex energy landscapes	163	(1)
Random fields and random bonds	164	(1)
Spin glasses and optimization by simulated annealing	165	(4)
Ageing in spin glasses and related systems	169	(1)
Vector spin glasses: developments and surprises	170	(1)
Models with mixed degrees of freedom: Si/Ge alloys, a case study	171	(1)
Sampling the free energy and entropy	172	(4)
Thermodynamic integration	172	(2)
Groundstate free energy determination	174	(1)
Estimation of intensive variables: the chemical potential	174	(1)
Lee--Kosterlitz method	175	(1)
Free energy from finite size dependence at Tc	175	(1)
Miscellaneous topics	176	(13)
Inhomogeneous systems: surfaces, interfaces, etc.	176	(6)
Other Monte Carlo schemes	182	(2)
Inverse Monte Carlo methods	184	(1)
Finite size effects: a review and summary	185	(1)
More about error estimation	186	(1)
Random number generators revisited	187	(2)
Summary and perspective	189	(5)
References	190	(4)
Off-lattice models	194	(57)
Fluids	194	(28)
NVT ensemble and the virial theorem	194	(3)
NpT ensemble	197	(4)
Grand canonical ensemble	201	(4)
Near critical coexistence: a case study	205	(2)
Subsystems: a case study	207	(5)
Gibbs ensemble	212	(3)
Widom particle insertion method and variants	215	(2)
Monte Carlo Phase Switch	217	(3)
Cluster algorithm for fluids	220	(2)
`Short range' interactions	222	(1)
Cutoffs	222	(1)
Verlet tables and cell structure	222	(1)
Minimum image convention	222	(1)
Mixed degrees of freedom reconsidered	223	(1)
Treatment of long range forces	223	(3)
Reaction field method	223	(1)
Ewald method	224	(1)
Fast multipole method	225	(1)
Absorbed monolayers	226	(1)
Smooth substrates	226	(1)
Periodic substrate potentials	226	(1)
Complex fluids	227	(4)
Application of the Liu-Luijten algorithm to a binary fluid mixture	230	(1)
Polymers: an introduction	231	(14)
Length scales and models	231	(6)
Asymmetric polymer mixtures: a case study	237	(3)
Applications: dynamics of polymer melts; thin adsorbed polymeric films	240	(5)
Configurational bias and `smart Monte Carlo'	245	(6)
References	248	(3)
Reweighting methods	251	(26)
Background	251	(3)
Distribution functions	251	(1)
Umbrella sampling	251	(3)
Single histogram method: the Ising model as a case study	254	(7)
Multi-histogram method	261	(1)
Broad histogram method	262	(1)
Transition matrix Monte Carlo	262	(1)
Multicanonical sampling	263	(5)
The multicanonical approach and its relationship to canonical sampling	263	(1)
Near first order transitions	264	(2)
Groundstates in complicated energy landscapes	266	(1)
Interface free energy estimation	267	(1)
A case study: the Casimir effect in critical systems	268	(2)
`Wang-Landau sampling'	270	(3)
A case study: evaporation/condensation transition of droplets	273	(4)
References	274	(3)
Quantum Monte Carlo methods	277	(38)
Introduction	277	(2)
Feynman path integral formulation	279	(9)
Off-lattice problems: low-temperature properties of crystals	279	(6)
Bose statistics and superfluidity	285	(1)
Path integral formulation for rotational degrees of freedom	286	(2)
Lattice problems	288	(19)
The Ising model in a transverse field	288	(2)
Anisotropic Heisenberg chain	290	(3)
Fermions on a lattice	293	(3)
An intermezzo: the minus sign problem	296	(2)
Spinless fermions revisited	298	(3)
Cluster methods for quantum lattice models	301	(1)
Continuous time simulations	302	(1)
Decoupled cell method	302	(1)
Handscomb's method	303	(1)
Wang-Landau sampling for quantum models	304	(2)
Fermion determinants	306	(1)
Monte Carlo methods for the study of groundstate properties	307	(4)
Variational Monte Carlo (VMC)	308	(1)
Green's function Monte Carlo methods (GFMC)	309	(2)
Concluding remarks	311	(4)
References	312	(3)
Monte Carlo renormalization group methods	315	(13)
Introduction to renormalization group theory	315	(4)
Real space renormalization group	319	(1)
Monte Carlo renormalization group	320	(8)
Large cell renormalization	320	(2)
Ma's method: finding critical exponents and the fixed point Hamiltonian	322	(1)
Swendsen's method	323	(2)
Location of phase boundaries	325	(1)
Dynamic problems: matching time-dependent correlation functions	326	(1)
Inverse Monte Carlo renormalization group transformations	327	(1)
References	327	(1)
Non-equilibrium and irreversible processes	328	(22)
Introduction and perspective	328	(1)
Driven diffusive systems (driven lattice gases)	328	(3)
Crystal growth	331	(2)
Domain growth	333	(3)
Polymer growth	336	(1)
Linear polymers	336	(1)
Gelation	336	(1)
Growth of structures and patterns	337	(5)
Eden model of cluster growth	337	(1)
Diffusion limited aggregation	338	(2)
Cluster--cluster aggregation	340	(1)
Cellular automata	340	(2)
Models for film growth	342	(5)
Background	342	(1)
Ballistic deposition	343	(1)
Sedimentation	343	(1)
Kinetic Monte Carlo and MBE growth	344	(3)
Transition path sampling	347	(1)
Outlook: variations on a theme	348	(2)
References	348	(2)
Lattice gauge models: a brief introduction	350	(13)
Introduction: gauge invariance and lattice gauge theory	350	(2)
Some technical matters	352	(1)
Results for Z(N) lattice gauge models	352	(1)
Compact U(1) gauge theory	353	(1)
SU(2) lattice gauge theory	354	(1)
Introduction: quantum chromodynamics (QCD) and phase transitions of nuclear matter	355	(2)
The deconfinement transition of QCD	357	(3)
Where are we now?	360	(3)
References	362	(1)
A brief review of other methods of computer simulation	363	(20)
Introduction	363	(1)
Molecular dynamics	363	(9)
Integration methods (microcanonical ensemble)	363	(4)
Other ensembles (constant temperature, constant pressure, etc.)	367	(3)
Non-equilibrium molecular dynamics	370	(1)
Hybrid methods (MD + MC)	370	(1)
Ab initio molecular dynamics	371	(1)
Quasi-classical spin dynamics	372	(3)
Langevin equations and variations (cell dynamics)	375	(1)
Micromagnetics	376	(1)
Dissipative particle dynamics (DPPD)	377	(1)
Lattice gas cellular automata	378	(1)
Lattice Boltzmann Equation	379	(1)
Multiscale simulation	379	(4)
References	381	(2)
Monte Carlo methods outside of physics	383	(10)
Commentary	383	(1)
Protein folding	383	(4)
Introduction	383	(1)
Generalized ensemble methods	384	(2)
Globular proteins: a case study	386	(1)
`Biologically inspired physics'	387	(1)
Mathematics/statistics	388	(1)
Sociophysics	388	(1)
Econophysics	388	(1)
`Traffic' simulations	389	(2)
Medicine	391	(2)
References	392	(1)
Outlook	393	(2)
Appendix: listing of programs mentioned in the text	395	(32)
Index	427