Statistical Mechanics: Algorithms and Computations

1 Monte Carlo methods......Page 14
1.1.1 Direct sampling......Page 16
1.1.2 Markov-chain sampling......Page 17
1.1.3 Historical origins......Page 22
1.1.4 Detailed balance......Page 28
1.1.5 The Metropolis algorithm......Page 34
1.1.6 A priori probabilities, triangle algorithm......Page 35
1.1.7 Perfect sampling with Markov chains......Page 37
1.2.1 Real random numbers......Page 40
1.2.2 Random integers, permutations, and combinations......Page 42
1.2.3 Finite distributions......Page 46
1.2.4 Continuous distributions and sample transformation......Page 48
1.2.5 Gaussians......Page 50
1.2.6 Random points in/on a sphere......Page 52
1.3.1 Sum of random variables, convolution......Page 57
1.3.2 Mean value and variance......Page 61
1.3.3 The central limit theorem......Page 65
1.3.4 Data analysis for independent variables......Page 68
1.3.5 Error estimates for Markov chains......Page 72
1.4.1 Ergodicity......Page 75
1.4.2 Importance sampling......Page 76
1.4.3 Monte Carlo quality control......Page 81
1.4.4 Stable distributions......Page 83
1.4.5 Minimum number of samples......Page 89
2 Hard disks and spheres......Page 93
2.1.1 Pair collisions and wall collisions......Page 96
2.1.2 Chaotic dynamics......Page 99
2.1.3 Observables......Page 100
2.1.4 Periodic boundary conditions......Page 103
2.2 Boltzmann’s statistical mechanics......Page 105
2.2.1 Direct disk sampling......Page 108
2.2.2 Partition function for hard disks......Page 110
2.2.3 Markov-chain hard-sphere algorithm......Page 113
2.2.4 Velocities: the Maxwell distribution......Page 116
2.2.5 Hydrodynamics: long-time tails......Page 118
2.3 Pressure and the Boltzmann distribution......Page 121
2.3.1 Bath-and-plate system......Page 122
2.3.2 Piston-and-plate system......Page 124
2.3.3 Ideal gas at constant pressure......Page 126
2.3.4 Constant-pressure simulation of hard spheres......Page 128
2.4.1 Grid/cell schemes......Page 132
2.4.2 Liquid–solid transitions......Page 133
2.5 Cluster algorithms......Page 135
2.5.1 Avalanches and independent sets......Page 136
2.5.2 Hard-sphere cluster algorithm......Page 138
3 Density matrices and path integrals......Page 144
3.1 Density matrices......Page 145
3.1.1 The quantum harmonic oscillator......Page 146
3.1.2 Free density matrix......Page 148
3.1.3 Density matrices for a box......Page 150
3.1.4 Density matrix in a rotating box......Page 152
3.2.1 High-temperature limit, convolution......Page 156
3.2.2 Harmonic oscillator (exact solution)......Page 158
3.2.3 Infinitesimal matrix products......Page 161
3.3 The Feynman path integral......Page 162
3.3.1 Naive path sampling......Page 163
3.3.2 Direct path sampling and the L´evy construction......Page 165
3.3.3 Periodic boundary conditions, paths in a box......Page 168
3.4 Pair density matrices......Page 172
3.4.1 Two quantum hard spheres......Page 173
3.4.2 Perfect pair action......Page 175
3.4.3 Many-particle density matrix......Page 180
3.5 Geometry of paths......Page 181
3.5.1 Paths in Fourier space......Page 182
3.5.2 Path maxima, correlation functions......Page 187
3.5.3 Classical random paths......Page 190
4 Bosons......Page 197
4.1.1 Single-particle density of states......Page 200
4.1.2 Trapped bosons (canonical ensemble)......Page 203
4.1.3 Trapped bosons (grand canonical ensemble)......Page 209
4.1.4 Large-N limit in the grand canonical ensemble......Page 213
4.1.5 Differences between ensembles—fluctuations......Page 218
4.1.6 Homogeneous Bose gas......Page 219
4.2.1 Bosonic density matrix......Page 222
4.2.2 Recursive counting of permutations......Page 225
4.2.3 Canonical partition function of ideal bosons......Page 226
4.2.4 Cycle-length distribution, condensate fraction......Page 230
4.2.5 Direct-sampling algorithm for ideal bosons......Page 232
4.2.6 Homogeneous Bose gas, winding numbers......Page 234
4.2.7 Interacting bosons......Page 237
5 Order and disorder in spin systems......Page 242
5.1 The Ising model—exact computations......Page 244
5.1.1 Listing spin configurations......Page 245
5.1.2 Thermodynamics, specific heat capacity, and magnetization......Page 247
5.1.3 Listing loop configurations......Page 249
5.1.4 Counting (not listing) loops in two dimensions......Page 253
5.1.5 Density of states from thermodynamics......Page 260
5.2.1 Local sampling methods......Page 262
5.2.2 Heat bath and perfect sampling......Page 265
5.2.3 Cluster algorithms......Page 267
5.3.1 The two-dimensional spin glass......Page 272
5.3.2 Liquids as Ising-spin-glass models......Page 275
6 Entropic forces......Page 279
6.1.1 Random clothes-pins......Page 282
6.1.2 The Asakura–Oosawa depletion interaction......Page 286
6.1.3 Binary mixtures......Page 290
6.2.1 Basic enumeration......Page 294
6.2.2 Breadth-first and depth-first enumeration......Page 297
6.2.3 Pfaffian dimer enumerations......Page 301
6.2.4 Monte Carlo algorithms for the monomer–dimer problem......Page 309
6.2.5 Monomer–dimer partition function......Page 312
7 Dynamic Monte Carlo methods......Page 320
7.1 Random sequential deposition......Page 322
7.1.1 Faster-than-the-clock algorithms......Page 323
7.2 Dynamic spin algorithms......Page 326
7.2.1 Spin-flips and dice throws......Page 327
7.2.2 Accelerated algorithms for discrete systems......Page 330
7.2.3 Futility......Page 332
7.3 Disks on the unit sphere......Page 334
7.3.1 Simulated annealing......Page 337
7.3.2 Asymptotic densities and paper-cutting......Page 340
7.3.3 Polydisperse disks and the glass transition......Page 343
7.3.4 Jamming and planar graphs......Page 344
Index......Page 352