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Contents xi

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 idealBose gas (density matrices) 209
4.2.1 Bosonic density matrix 209
4.2.2 Recursivecounting ofpermutations 212
4.2.3 Canonical partitionfunction ofideal bosons 213
4.2.4 Cycle-length distribution, condensate fraction 217
4.2.5 Direct-sampling algorithm forideal 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 mag-
netization 234
5.1.3 Listing loopconfigurations 236
5.1.4 Counting (not listing)loops in two dimensions 240
5.1.5 Densityof states fromthermodynamics 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 continuummodelsand mixtures 269
6.1.1 Randomclothes-pins 269
6.1.2 The Asakura–Oosawadepletioninteraction 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 forthe monomer–dimer
problem 296
6.2.5 Monomer–dimer partitionfunction 299
Exercises 303
References 305

7 Dynamic Monte Carlo methods 307