A Guide to Monte Carlo Simulations in Statistical Physics

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
1.1 WHAT IS A MONTE CARLO SIMULATION?......Page 19
1.2 WHAT PROBLEMS CAN WE SOLVE WITH IT?......Page 20
1.3.2 Statistical and other errors......Page 21
1.5 HOW DO SIMULATIONS RELATE TO THEORY AND EXPERIMENT?......Page 22
REFERENCES......Page 24
2.1.1.1 Partition function......Page 25
2.1.1.2 Free energy, internal energy, and entropy......Page 27
2.1.1.3 Thermodynamic potentials and corresponding ensembles......Page 28
2.1.1.4 Fluctuations......Page 30
2.1.2 Phase transitions......Page 31
2.1.2.2 Correlation function......Page 32
2.1.2.3 First order vs. second order......Page 33
2.1.2.4 Phase diagrams......Page 34
2.1.2.5 Critical behavior and exponents......Page 35
2.1.2.6 Universality and scaling......Page 37
2.1.2.7 Multicritical phenomena......Page 40
2.1.2.8 Landau theory......Page 42
2.1.3 Ergodicity and broken symmetry......Page 43
2.1.4 Fluctuations and the Ginzburg criterion......Page 44
2.1.5 A standard exercise: the ferromagnetic Isingmodel......Page 45
2.2.1 Basic notions......Page 46
2.2.2 Special probability distributions and the central limit theorem......Page 48
2.2.3 Statistical errors......Page 49
2.2.4 Markov chains andmaster equations......Page 50
2.2.5.1 Background......Page 52
2.2.5.4 Shift register algorithms......Page 53
2.2.5.5 Lagged Fibonacci generators......Page 54
2.2.5.6 Tests for quality......Page 55
2.3.1 Physical applications of master equations......Page 57
2.3.2 Conservation laws and their consequences......Page 59
2.3.3 Critical slowing down at phase transitions......Page 62
2.3.4 Transport coefficients......Page 63
REFERENCES......Page 64
3.2.1 Simplemethods......Page 66
3.2.2 Intelligentmethods......Page 68
3.3 BOUNDARY VALUE PROBLEMS......Page 69
3.4 SIMULATION OF RADIOACTIVE DECAY......Page 71
3.5.1 Neutron transport......Page 72
3.5.2 Fluid flow......Page 73
3.6.1 Site percolation......Page 74
3.6.2 Cluster counting: the Hoshen-Kopelman algorithm......Page 77
3.7 FINDING THE GROUNDSTATE OF A HAMILTONIAN......Page 78
3.8.1 Introduction......Page 79
3.8.2 Random walks......Page 80
3.8.3 Self-avoiding walks......Page 81
3.8.4 Growing walks and othermodels......Page 83
REFERENCES......Page 84
4.1 INTRODUCTION......Page 86
4.2 THE SIMPLEST CASE: SINGLE SPIN-FLIP SAMPLING FOR THE SIMPLE ISING MODEL......Page 87
4.2.1 Algorithm......Page 88
4.2.2.2 Screw periodic boundary conditions......Page 92
4.2.2.5 Free edge boundary conditions......Page 93
4.2.2.7 Hyperspherical boundary conditions......Page 94
4.2.3.2 Finite size scaling and critical exponents......Page 95
4.2.3.3 Finite size scaling and first order transitions......Page 100
4.2.3.5 Field mixing......Page 104
4.2.3.6 Finite size effects in simulations of interfaces......Page 105
4.2.4 Finite sampling time effects......Page 108
4.2.4.1 Statistical error......Page 109
4.2.4.2 Biased sampling error: Ising criticality as an example......Page 111
4.2.4.3 Relaxation effects......Page 113
4.2.4.4 Back to finite size effects again: self-averaging......Page 115
4.2.5 Critical relaxation......Page 116
4.2.5.1 Non-linear relaxation......Page 117
4.2.5.2 Linear relaxation......Page 119
4.2.5.4 Dynamic finite size scaling......Page 121
4.3.1 Ising models with competing interactions......Page 123
4.3.2 q-state Pottsmodels......Page 127
4.3.3 Baxter and Baxter^Wumodels......Page 128
4.3.4 Clock models......Page 129
4.3.5 Ising spin glassmodels......Page 131
4.3.6 Complex fluidmodels......Page 132
4.4.2 Phase separation......Page 133
4.4.3 Diffusion......Page 135
4.5.1 Demon algorithm......Page 138
4.5.3 Q2R......Page 139
4.7.1 Background......Page 140
4.7.2 Fixed bond length methods......Page 141
4.7.3 Bond fluctuationmethod......Page 143
4.7.4 Enhanced sampling using a fourth dimension......Page 144
4.7.5 The ‘wormhole algorithm’ – anothermethod to equilibrate dense polymeric systems......Page 145
4.7.6 Polymers in solutions of variable quality: theta point, collapse transition, unmixing......Page 146
4.7.7 Equilibrium polymers: a case study......Page 148
4.8 SOME ADVICE......Page 151
REFERENCES......Page 152
5.1.1 Fortuin-Kasteleyn theorem......Page 156
5.1.2 Swendsen-Wang method......Page 157
5.1.3 Wolff method......Page 160
5.1.5 Invaded cluster algorithm......Page 161
5.1.6 Probability changing cluster algorithm......Page 162
5.2.2 Multispin coding......Page 163
5.2.3 N-fold way and extensions......Page 164
5.2.6 Monte Carlo on vector computers......Page 167
5.2.7 Monte Carlo on parallel computers......Page 168
5.3.1 Introduction......Page 169
5.3.2 Simple spin-flip method......Page 170
5.3.3 Heatbath method......Page 172
5.3.5 Over-relaxationmethods......Page 173
5.3.6 Wolff embedding trick and cluster flipping......Page 174
5.3.8 Monte Carlo dynamics vs. equation of motion dynamics......Page 175
5.3.9 Topological excitations and solitons......Page 176
5.4.1 General comments: averaging in random systems......Page 178
5.4.3 Random fields and random bonds......Page 183
5.4.4 Spin glasses and optimization by simulated annealing......Page 184
5.4.5 Ageing in spin glasses and related systems......Page 189
5.4.6 Vector spin glasses: developments and surprises......Page 190
5.5 MODELS WITH MIXED DEGREES OF FREEDOM: Si/Ge ALLOYS, A CASE STUDY......Page 191
5.6.1 Thermodynamic integration......Page 192
5.6.2 Groundstate free energy determination......Page 194
5.6.4 Lee-Kosterlitzmethod......Page 195
5.7.1 Inhomogeneous systems: surfaces, interfaces, etc.......Page 196
5.7.2.1 Damage spreading......Page 202
5.7.2.3 Simulations at more than one length scale......Page 203
5.7.3 Inverse and reverse Monte Carlo methods......Page 204
5.7.4 Finite size effects: a review and summary......Page 205
5.7.5 More about error estimation......Page 206
5.7.6 Random number generators revisited......Page 208
REFERENCES......Page 211
6.1.1 NVT ensemble and the virial theorem......Page 215
6.1.2 NpT ensemble......Page 218
6.1.3 Grand canonical ensemble......Page 222
6.1.4 Near critical coexistence: a case study......Page 226
6.1.5 Subsystems: a case study......Page 228
6.1.6 Gibbs ensemble......Page 233
6.1.7 Widom particle insertionmethod and variants......Page 236
6.1.8 Monte Carlo Phase Switch......Page 238
6.1.9 Cluster algorithm for fluids......Page 242
6.2.2 Verlet tables and cell structure......Page 243
6.3.1 Reaction fieldmethod......Page 244
6.3.2 Ewald method......Page 245
6.3.3 Fastmultipolemethod......Page 246
6.4.2 Periodic substrate potentials......Page 247
6.5 COMPLEX FLUIDS......Page 249
6.5.1 Application of the Liu-Luijten algorithm to a binary fluid mixture......Page 251
6.6.1 Length scales and models......Page 252
6.6.2 Asymmetric polymer mixtures: a case study......Page 259
6.6.3 Applications: dynamics of polymer melts; thin adsorbed polymeric films......Page 263
6.6.4 Polymer melts: speeding up bond fluctuation model simulations......Page 266
6.7 CONFIGURATIONAL BIAS AND ‘SMART MONTE CARLO’......Page 268
REFERENCES......Page 271
7.1.2 Umbrella sampling......Page 275
7.2 SINGLE HISTOGRAM METHOD: THE ISING MODEL AS A CASE STUDY......Page 278
7.3 MULTIHISTOGRAM METHOD......Page 285
7.5 TRANSITION MATRIX MONTE CARLO......Page 286
7.6.1 The multicanonical approach and its relationship to canonical sampling......Page 287
7.6.2 Near first order transitions......Page 288
7.6.3 Groundstates in complicated energy landscapes......Page 290
7.6.4 Interface free energy estimation......Page 291
7.7 A CASE STUDY: THE CASIMIR EFFECT IN CRITICAL SYSTEMS......Page 292
7.8.1 Basic algorithm......Page 294
7.8.4 Back to numerical integration......Page 297
7.9 A CASE STUDY: EVAPORATION/CONDENSATION TRANSITION OF DROPLETS......Page 299
REFERENCES......Page 300
8.1 INTRODUCTION......Page 303
8.2.1 Off-lattice problems: low-temperature properties of crystals......Page 305
8.2.2 Bose statistics and superfluidity......Page 311
8.2.3 Path integral formulation for rotational degrees of freedom......Page 312
8.3.1 The Isingmodel in a transverse field......Page 315
8.3.2 AnisotropicHeisenberg chain......Page 316
8.3.3 Fermions on a lattice......Page 320
8.3.4 An intermezzo: the minus sign problem......Page 322
8.3.5 Spinless fermions revisited......Page 324
8.3.7 Continuous time simulations......Page 328
8.3.8 Decoupled cell method......Page 329
8.3.9 Handscomb\'s method......Page 330
8.3.10 Wang-Landau sampling for quantum models......Page 331
8.3.11 Fermion determinants......Page 332
8.4.1 Variational Monte Carlo (VMC)......Page 334
8.4.2 Green\'s function Monte Carlo methods (GFMC)......Page 336
8.5 CONCLUDING REMARKS......Page 338
REFERENCES......Page 339
9.1 INTRODUCTION TO RENORMALIZATION GROUP THEORY......Page 342
9.2 REAL SPACE RENORMALIZATION GROUP......Page 346
9.3.1 Large cell renormalization......Page 347
9.3.2 Ma’smethod: f|nding critical exponents and the fixed point Hamiltonian......Page 349
9.3.3 Swendsen\'s method......Page 350
9.3.4.1 Critical points......Page 352
9.3.5 Dynamic problems: matching time-dependent correlation functions......Page 353
REFERENCES......Page 354
10.2 DRIVEN DIFFUSIVE SYSTEMS (DRIVEN LATTICE GASES)......Page 356
10.3 CRYSTAL GROWTH......Page 359
10.4 DOMAIN GROWTH......Page 362
10.5.2 Gelation......Page 365
10.6.2 Diffusion limited aggregation......Page 367
10.6.2.2 Off-lattice DLA......Page 368
10.6.4 Cellular automata......Page 370
10.7.1 Background......Page 371
10.7.2 Ballistic deposition......Page 372
10.7.3 Sedimentation......Page 373
10.7.4 Kinetic Monte Carlo and MBE growth......Page 374
10.8 TRANSITION PATH SAMPLING......Page 376
10.9 FORCED POLYMER PORE TRANSLOCATION: A CASE STUDY......Page 377
REFERENCES......Page 380
11.1 INTRODUCTION: GAUGE INVARIANCE AND LATTICE GAUGE THEORY......Page 383
11.3 RESULTS FOR Z(N) LATTICE GAUGE MODELS......Page 385
11.4 COMPACT U(1) GAUGE THEORY......Page 386
11.5 SU(2) LATTICE GAUGE THEORY......Page 387
11.6 INTRODUCTION: QUANTUM CHROMODYNAMICS (QCD) AND PHASE TRANSITIONS OF NUCLEAR MATTER......Page 388
11.7 THE DECONFINEMENT TRANSITION OF QCD......Page 390
11.8 TOWARDS QUANTITATIVE PREDICTIONS......Page 393
REFERENCES......Page 395
12.2.1 Integrationmethods (microcanonical ensemble)......Page 397
12.2.2 Other ensembles (constant temperature, constant pressure, etc.)......Page 401
12.2.4 Hybrid methods (MD + MC)......Page 404
12.2.5 Ab initio molecular dynamics......Page 405
12.3 QUASI-CLASSICAL SPIN DYNAMICS......Page 406
12.4 LANGEVIN EQUATIONS AND VARIATIONS (CELL DYNAMICS)......Page 410
12.6 DISSIPATIVE PARTICLE DYNAMICS (DPD)......Page 411
12.8 LATTICE BOLTZMANN EQUATION......Page 413
12.9 MULTISCALE SIMULATION......Page 414
REFERENCES......Page 416
13.2 ASTROPHYSICS......Page 419
13.3 MATERIALS SCIENCE......Page 420
13.4 CHEMISTRY......Page 421
13.5.1 Commentary and perspective......Page 423
13.5.2 Lattice proteins......Page 424
13.5.3 Cell sorting......Page 425
13.6 BIOLOGY......Page 427
13.8 SOCIOPHYSICS......Page 428
13.9 ECONOPHYSICS......Page 429
13.10 \'TRAFFIC\' SIMULATIONS......Page 430
13.11 MEDICINE......Page 431
13.12 NETWORKS: WHAT CONNECTIONS REALLY MATTER?......Page 432
13.13 FINANCE......Page 433
REFERENCES......Page 434
14.1 INTRODUCTION......Page 437
14.2.1 Introduction......Page 438
14.2.3 Generalized ensemble methods......Page 439
14.2.4 Globular proteins: a case study......Page 441
14.2.5 Simulations of membrane proteins......Page 442
14.3 MONTE CARLO SIMULATIONS OF CARBOHYDRATES......Page 444
14.4 DETERMINING MACROMOLECULAR STRUCTURES......Page 445
REFERENCES......Page 446
15 Outlook......Page 448
Appendix: listing of programs mentioned in the text......Page 451
Index......Page 483