A GUIDE TO MONTE CARLO SIMULATIONS IN STATISTICAL PHYSICS

By F. JOANY

CONTENTS PAGE 1 INTRODUCTION 1 1 WHAT IS A MONTE CARLO SIMULATION? 1 2 WHAT PROBLEMS CAN WE SOLVE WITH IT? 1 3 WHAT DIF?CULTIES 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 COEF?CIENTS 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 TRANSPORT PROPERTIES 3 5 1 NEUTRON TRANSPORT 3 5 2 FLUID ?OW 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 4 IMPORTANCE SAMPLING MONTE CARLO METHODS 4 1 INTRODUCTION 4 2 THE SIMPLEST CASE: SINGLE SPIN-?IP 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 ?UID 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 ?UCTUATION 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 8 SOME ADVICE 5 MORE ON IMPORTANCE SAMPLING MONTE CARLO METHODS FOR LATTICE SYSTEMS 5 1 CLUSTER ?IPPING 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-?IP 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 ?IPPING 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 ?ELDS 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 ?NITE 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 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 ?UIDS 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 ?ELD 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 ?UIDS 6 5 1 APPLICATION OF THE LIU-LUIJTEN ALGORITHM TO A BINARY ?UID 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 ?LMS 6 7 CON?GURATIONAL BIAS AND ‘SMART MONTE CARLO’ 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 ?RST 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 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 SUPER?UIDITY 8 2 3 PATH INTEGRAL FORMULATION FOR ROTATIONAL DEGREES OF FREEDOM 8 3 LATTICE PROBLEMS 8 3 1 THE ISING MODEL IN A TRANSVERSE ?ELD 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 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: ?NDING CRITICAL EXPONENTS AND THE ?XED 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 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 ?LM 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 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 DECON?NEMENT TRANSITION OF QCD 11 8 WHERE ARE WE NOW? 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 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 ‘TRAF?C’ SIMULATIONS 13 8 MEDICINE 14 OUTLOOK