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Contents

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 Historicalorigins 9
1.1.4 Detailed balance 15
1.1.5 The Metropolis algorithm 21
1.1.6 A priori probabilities, trianglealgorithm 22
1.1.7 Perfect sampling with Markov chains 24
1.2 Basic sampling 27
1.2.1 Real randomnumbers 27
1.2.2 Random integers, permutations, and combinations 29
1.2.3 Finite distributions 33
1.2.4 Continuous distributions and sampletransformation 35
1.2.5 Gaussians 37
1.2.6 Randompoints in/on a sphere 39
1.3 Statistical data analysis 44
1.3.1 Sum ofrandom variables, convolution 44
1.3.2 Mean valueand variance 48
1.3.3 The centrallimit theorem 52
1.3.4 Data analysis for independent variables 55
1.3.5 Errorestimates 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 Minimumnumber ofsamples 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