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A Guide to Monte Carlo Simulations in Statistical Physics ebook代购

A Guide to Monte Carlo Simulations in Statistical Physics ebook代购

Contents
page
Preface xii
1 Introduction 1
1.1 What is a Monte Carlo simulation? 1
1.2 What problems can we solve with it? 2
1.3 What difficulties will we encounter? 3
1.3.1 Limited computer time and memory 3
1.3.2 Statistical and other errors 3
1.4 What strategy should we follow in approaching a problem? 4
1.5 How do simulations relate to theory and experiment? 4
1.6 Perspective 6
2 Some necessary background 7
2.1 Thermodynamics and statistical mechanics: a quick reminder 7
2.1.1 Basic notions 7
2.1.2 Phase transitions 13
2.1.3 Ergodicity and broken symmetry 24
2.1.4 Fluctuations and the Ginzburg criterion 25
2.1.5 A standard exercise: the ferromagnetic Ising model 25
2.2 Probability theory 27
2.2.1 Basic notions 27
2.2.2 Special probability distributions and the central limit theorem 29
2.2.3 Statistical errors 30
2.2.4 Markovch ains and master equations 31
2.2.5 The ‘art’ of random number generation 32
2.3 Non-equilibrium and dynamics: some introductory comments 39
2.3.1 Physical applications of master equations 39
2.3.2 Conservation laws and their consequences 40
2.3.3 Critical slowing down at phase transitions 43
2.3.4 Transport coefficients 45
2.3.5 Concluding comments 45
References 45
3 Simple sampling Monte Carlo methods 48
3.1 Introduction 48
3.2 Comparisons of methods for numerical integration of given
functions 48
v
3.2.1 Simple methods 48
3.2.2 Intelligent methods 50
3.3 Boundary value problems 51
3.4 Simulation of radioactive decay 53
3.5 Simulation of transport properties 54
3.5.1 Neutron transport 54
3.5.2 Fluid flow 55
3.6 The percolation problem 56
3.6.1 Site percolation 56
3.6.2 Cluster counting: the Hoshen–Kopelman algorithm 59
3.6.3 Other percolation models 60
3.7 Finding the groundstate of a Hamiltonian 60
3.8 Generation of ‘random’ walks 61
3.8.1 Introduction 61
3.8.2 Random walks 62
3.8.3 Self-avoiding walks 63
3.8.4 Growing walks and other models 65
3.9 Final remarks 66
References 66
4 Importance sampling Monte Carlo methods 68
4.1 Introduction 68
4.2 The simplest case: single spin-flip sampling for the simple Ising
model 69
4.2.1 Algorithm 70
4.2.2 Boundary conditions 74
4.2.3 Finite size effects 77
4.2.4 Finite sampling time effects 90
4.2.5 Critical relaxation 98
4.3 Other discrete variable models 105
4.3.1 Ising models with competing interactions 105
4.3.2 q-state Potts models 109
4.3.3 Baxter and Baxter–Wu models 110
4.3.4 Clock models 111
4.3.5 Ising spin glass models 113
4.3.6 Complex fluid models 114
4.4 Spin-exchange sampling 115
4.4.1 Constant magnetization simulations 115
4.4.2 Phase separation 115
4.4.3 Diffusion 117
4.4.4 Hydrodynamic slowing down 120
4.5 Microcanonical methods 120
4.5.1 Demon algorithm 120
4.5.2 Dynamic ensemble 121
4.5.3 Q2R 121
4.6 General remarks, choice of ensemble 122
vi Contents
4.7 Statics and dynamics of polymer models on lattices 122
4.7.1 Background 122
4.7.2 Fixed bond length methods 123
4.7.3 Bond fluctuation method 124
4.7.4 Enhanced sampling using a fourth dimension 125
4.7.5 The ‘wormhole algorithm’ – another method to equilibrate
dense polymeric systems 127
4.7.6 Polymers in solutions of variable quality: -point, collapse
transition, unmixing 127
4.7.7 Equilibrium polymers: a case study 130
4.8 Some advice 133
References 134
5 More on importance sampling Monte Carlo methods for
lattice systems 137
5.1 Cluster flipping methods 137
5.1.1 Fortuin–Kasteleyn theorem 137
5.1.2 Swendsen–Wang method 138
5.1.3 Wolff method 141
5.1.4 ‘Improved estimators’ 142
5.1.5 Invaded cluster algorithm 142


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