Statistical Mechanics

This book discusses the computational approach in modern statistical physics in a clear yet accessible way, and works out its intimate relations with other approaches in theoretical physics. Individual chapters focus on subjects as diverse as the hard sphere liquid, classical spin models, single quantum particles and Bose-Einstein condensation. They contain in-depth discussions...

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This book discusses the computational approach in modern statistical physics in a clear yet accessible way, and works out its intimate relations with other approaches in theoretical physics. Individual chapters focus on subjects as diverse as the hard sphere liquid, classical spin models, single quantum particles and Bose-Einstein condensation. They contain in-depth discussions of algorithms ranging from basic enumeration methods to modern Monte Carlo techniques. The emphasis is on orientation. Discussions of implementation details are kept to a minimum. The book heavily relies on illustrations, tables and concise printed algorithms to convey key information: all the material remains easily accessible. The book is fully self-contained: graphs and tables can be readily reproduced by programming at most a few dozen lines of computer code. Most sections lead from an elementary discussion to the rich and difficult problems of contemporary computational and statistical physics, and will be of interest to a wide range of students, teachers and researchers in physics and the neighboring sciences.An accompanying CD allows to incorporate the layout material (illustrations, tables, schematic programs) into the reader's own presentations.

1 Monte Carlo methods 1
1.1 Popular games in Monaco 3
1.1.1 Direct sampling 3
1.1.2 Markov-chain sampling 4
1.1.3 Historical origins 9
1.1.4 Detailed balance 15
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1 Monte Carlo methods 1
1.1 Popular games in Monaco 3
1.1.1 Direct sampling 3
1.1.2 Markov-chain sampling 4
1.1.3 Historical origins 9
1.1.4 Detailed balance 15
1.1.5 The Metropolis algorithm 21
1.1.6 A priori probabilities, triangle algorithm 22
1.1.7 Perfect sampling with Markov chains 24
1.2 Basic sampling 27
1.2.1 Real random numbers 27
1.2.2 Random integers, permutations, and combinations 29
1.2.3 Finite distributions 33
1.2.4 Continuous distributions and sample transformation 35
1.2.5 Gaussians 37
1.2.6 Random points in/on a sphere 39
1.3 Statistical data analysis 44
1.3.1 Sum of random variables, convolution 44
1.3.2 Mean value and variance 48
1.3.3 The central limit theorem 52
1.3.4 Data analysis for independent variables 55
1.3.5 Error estimates for Markov chains 59
1.4 Computing 62
1.4.1 Ergodicity 62
1.4.2 Importance sampling 63
1.4.3 Monte Carlo quality control 68
1.4.4 Stable distributions 70
1.4.5 Minimum number of samples 76
Exercises 77
References 79
2 Hard disks and spheres 81
2.1 Newtonian deterministic mechanics 83
2.1.1 Pair collisions and wall collisions 83
2.1.2 Chaotic dynamics 86
2.1.3 Observables 87
2.1.4 Periodic boundary conditions 90
2.2 Boltzmann’s statistical mechanics 92
2.2.1 Direct disk sampling 95
2.2.2 Partition function for hard disks 97
2.2.3 Markov-chain hard-sphere algorithm 100
2.2.4 Velocities: the Maxwell distribution 103
2.2.5 Hydrodynamics: long-time tails 105
2.3 Pressure and the Boltzmann distribution 108
2.3.1 Bath-and-plate system 109
2.3.2 Piston-and-plate system 111
2.3.3 Ideal gas at constant pressure 113
2.3.4 Constant-pressure simulation of hard spheres 115
2.4 Large hard-sphere systems 119
2.4.1 Grid/cell schemes 119
2.4.2 Liquid–solid transitions 120
2.5 Cluster algorithms 122
2.5.1 Avalanches and independent sets 123
2.5.2 Hard-sphere cluster algorithm 125
Exercises 128
References 130
3 Density matrices and path integrals 131
3.1 Density matrices 133
3.1.1 The quantum harmonic oscillator 133
3.1.2 Free density matrix 135
3.1.3 Density matrices for a box 137
3.1.4 Density matrix in a rotating box 139
3.2 Matrix squaring 143
3.2.1 High-temperature limit, convolution 143
3.2.2 Harmonic oscillator (exact solution) 145
3.2.3 Infinitesimal matrix products 148
3.3 The Feynman path integral 149
3.3.1 Naive path sampling 150
3.3.2 Direct path sampling and the L´evy construction 152
3.3.3 Periodic boundary conditions, paths in a box 155
3.4 Pair density matrices 159
3.4.1 Two quantum hard spheres 160
3.4.2 Perfect pair action 162
3.4.3 Many-particle density matrix 167
3.5 Geometry of paths 168
3.5.1 Paths in Fourier space 169
3.5.2 Path maxima, correlation functions 174
3.5.3 Classical random paths 177
Exercises 182
References 184
4 Bosons 185
4.1 Ideal bosons (energy levels) 187
4.1.1 Single-particle density of states 187
4.1.2 Trapped bosons (canonical ensemble) 190
4.1.3 Trapped bosons (grand canonical ensemble) 196
4.1.4 Large-N limit in the grand canonical ensemble 200
4.1.5 Differences between ensembles—fluctuations 205
4.1.6 Homogeneous Bose gas 206
4.2 The ideal Bose gas (density matrices) 209
4.2.1 Bosonic density matrix 209
4.2.2 Recursive counting of permutations 212
4.2.3 Canonical partition function of ideal bosons 213
4.2.4 Cycle-length distribution, condensate fraction 217
4.2.5 Direct-sampling algorithm for ideal bosons 219
4.2.6 Homogeneous Bose gas, winding numbers 221
4.2.7 Interacting bosons 224
Exercises 225
References 227
5 Order and disorder in spin systems 229
5.1 The Ising model—exact computations 231
5.1.1 Listing spin configurations 232
5.1.2 Thermodynamics, specific heat capacity, and magnetization
234
5.1.3 Listing loop configurations 236
5.1.4 Counting (not listing) loops in two dimensions 240
5.1.5 Density of states from thermodynamics 247
5.2 The Ising model—Monte Carlo algorithms 249
5.2.1 Local sampling methods 249
5.2.2 Heat bath and perfect sampling 252
5.2.3 Cluster algorithms 254
5.3 Generalized Ising models 259
5.3.1 The two-dimensional spin glass 259
5.3.2 Liquids as Ising-spin-glass models 262
Exercises 264
References 266
6 Entropic forces 267
6.1 Entropic continuum models and mixtures 269
6.1.1 Random clothes-pins 269
6.1.2 The Asakura–Oosawa depletion interaction 273
6.1.3 Binary mixtures 277
6.2 Entropic lattice model: dimers 281
6.2.1 Basic enumeration 281
6.2.2 Breadth-first and depth-first enumeration 284
6.2.3 Pfaffian dimer enumerations 288
6.2.4 Monte Carlo algorithms for the monomer–dimer
problem 296
6.2.5 Monomer–dimer partition function 299
Exercises 303
References 305
7 Dynamic Monte Carlo methods 307
7.1 Random sequential deposition 309
7.1.1 Faster-than-the-clock algorithms 310
7.2 Dynamic spin algorithms 313
7.2.1 Spin-flips and dice throws 314
7.2.2 Accelerated algorithms for discrete systems 317
7.2.3 Futility 319
7.3 Disks on the unit sphere 321
7.3.1 Simulated annealing 324
7.3.2 Asymptotic densities and paper-cutting 327
7.3.3 Polydisperse disks and the glass transition 330
7.3.4 Jamming and planar graphs 331
Exercises 333
References 335
Acknowledgements 337
Index 339
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