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2.2.2 Partitionfunctionforhard 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 quantumharmonic oscillator 133
3.1.2 Free density matrix 135
3.1.3 Density matrices fora 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-particledensity matrix 167
3.5 Geometryofpaths 168
3.5.1 Paths in Fourier space 169
3.5.2 Path maxima, correlationfunctions 174
3.5.3 Classical randompaths 177
Exercises 182
References 184

4 Bosons 185
4.1 Ideal bosons (energylevels) 187
4.1.1 Single-particledensityof states 187
4.1.2 Trapped bosons (canonical ensemble) 190
4.1.3 Trapped bosons (grand canonical ensemble) 196