Preface |
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xiii | |
1 Introduction |
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1 | (6) |
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1.1 What is a Monte Carlo simulation? |
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1 | (1) |
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1.2 What problems can we solve with it? |
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2 | (1) |
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1.3 What difficulties will we encounter? |
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3 | (1) |
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1.3.1 Limited computer time and memory |
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3 | (1) |
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1.3.2 Statistical and other errors |
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3 | (1) |
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1.4 What strategy should we follow in approaching a problem? |
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4 | (1) |
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1.5 How do simulations relate to theory and experiment? |
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4 | (2) |
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6 | (1) |
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6 | (1) |
2 Some necessary background |
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7 | (41) |
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2.1 Thermodynamics and statistical mechanics: a quick reminder |
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7 | (21) |
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7 | (6) |
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13 | (12) |
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2.1.3 Ergodicity and broken symmetry |
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25 | (1) |
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2.1.4 Fluctuations and the Ginzburg criterion |
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26 | (1) |
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2.1.5 A standard exercise: the ferromagnetic Ising model |
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27 | (1) |
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28 | (11) |
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28 | (2) |
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2.2.2 Special probability distributions and the central limit theorem |
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30 | (1) |
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31 | (1) |
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2.2.4 Markov chains and master equations |
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32 | (2) |
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2.2.5 The 'art' of random number generation |
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34 | (5) |
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2.3 Non-equilibrium and dynamics: some introductory comments |
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39 | (7) |
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2.3.1 Physical applications of master equations |
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39 | (2) |
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2.3.2 Conservation laws and their consequences |
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41 | (3) |
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2.3.3 Critical slowing down at phase transitions |
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44 | (1) |
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2.3.4 Transport coefficients |
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45 | (1) |
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2.3.5 Concluding comments |
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46 | (1) |
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46 | (2) |
3 Simple sampling Monte Carlo methods |
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48 | (20) |
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48 | (1) |
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3.2 Comparisons of methods for numerical integration of given functions |
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48 | (3) |
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48 | (2) |
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3.2.2 Intelligent methods |
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50 | (1) |
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3.3 Boundary value problems |
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51 | (2) |
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3.4 Simulation of radioactive decay |
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53 | (1) |
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3.5 Simulation of transport properties |
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54 | (2) |
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54 | (1) |
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55 | (1) |
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3.6 The percolation problem |
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56 | (4) |
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56 | (3) |
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3.6.2 Cluster counting: the Hoshen-Kopelman algorithm |
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59 | (1) |
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3.6.3 Other percolation models |
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60 | (1) |
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3.7 Finding the groundstate of a Hamiltonian |
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60 | (1) |
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3.8 Generation of 'random' walks |
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61 | (5) |
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61 | (1) |
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62 | (1) |
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3.8.3 Self-avoiding walks |
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63 | (2) |
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3.8.4 Growing walks and other models |
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65 | (1) |
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66 | (1) |
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66 | (2) |
4 Importance sampling Monte Carlo methods |
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68 | (70) |
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68 | (1) |
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4.2 The simplest case: single spin-flip sampling for the simple Ising model |
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69 | (36) |
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70 | (4) |
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4.2.2 Boundary conditions |
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74 | (3) |
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4.2.3 Finite size effects |
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77 | (13) |
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4.2.4 Finite sampling time effects |
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90 | (8) |
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4.2.5 Critical relaxation |
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98 | (7) |
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4.3 Other discrete variable models |
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105 | (10) |
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4.3.1 Ising models with competing interactions |
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105 | (4) |
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4.3.2 q-state Potts models |
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109 | (1) |
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4.3.3 Baxter and Baxter-Wu models |
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110 | (1) |
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111 | (2) |
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4.3.5 Ising spin glass models |
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113 | (1) |
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4.3.6 Complex fluid models |
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114 | (1) |
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4.4 Spin-exchange sampling |
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115 | (5) |
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4.4.1 Constant magnetization simulations |
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115 | (1) |
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115 | (2) |
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117 | (3) |
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4.4.4 Hydrodynamic slowing down |
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120 | (1) |
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4.5 Microcanonical methods |
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120 | (2) |
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120 | (1) |
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121 | (1) |
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121 | (1) |
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4.6 General remarks, choice of ensemble |
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122 | (1) |
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4.7 Statics and dynamics of polymer models on lattices |
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122 | (11) |
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122 | (1) |
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4.7.2 Fixed bond length methods |
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123 | (2) |
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4.7.3 Bond fluctuation method |
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125 | (1) |
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4.7.4 Enhanced sampling using a fourth dimension |
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126 | (1) |
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4.7.5 The 'wormhole algorithm' - another method to equilibrate dense polymeric systems |
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127 | (1) |
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4.7.6 Polymers in solutions of variable quality: 0-point, collapse transition, unmixing |
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128 | (2) |
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4.7.7 Equilibrium polymers: a case study |
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130 | (3) |
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133 | (1) |
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134 | (4) |
5 More on importance sampling Monte Carlo methods for lattice systems |
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138 | (59) |
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5.1 Cluster flipping methods |
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138 | (7) |
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5.1.1 Fortuin-Kasteleyn theorem |
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138 | (1) |
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5.1.2 Swendsen-Wang method |
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139 | (3) |
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142 | (1) |
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5.1.4 'Improved estimators' |
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143 | (1) |
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5.1.5 Invaded cluster algorithm |
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143 | (1) |
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5.1.6 Probability changing cluster algorithm |
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144 | (1) |
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5.2 Specialized computational techniques |
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145 | (6) |
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5.2.1 Expanded ensemble methods |
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145 | (1) |
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145 | (1) |
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5.2.3 N-fold way and extensions |
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146 | (3) |
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149 | (1) |
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5.2.5 Multigrid algorithms |
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149 | (1) |
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5.2.6 Monte Carlo on vector computers |
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149 | (1) |
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5.2.7 Monte Carlo on parallel computers |
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150 | (1) |
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5.3 Classical spin models |
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151 | (9) |
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151 | (1) |
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5.3.2 Simple spin-flip method |
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152 | (2) |
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154 | (1) |
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5.3.4 Low temperature techniques |
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155 | (1) |
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5.3.5 Over-relaxation methods |
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155 | (1) |
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5.3.6 Wolff embedding trick and cluster flipping |
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156 | (1) |
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157 | (1) |
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5.3.8 Monte Carlo dynamics vs. equation of motion dynamics |
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157 | (1) |
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5.3.9 Topological excitations and solitons |
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158 | (2) |
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5.4 Systems with quenched randomness |
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160 | (13) |
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5.4.1 General comments: averaging in random systems |
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160 | (5) |
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5.4.2 Parallel tempering: a general method to better equilibrate systems with complex energy landscapes |
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165 | (1) |
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5.4.3 Random fields and random bonds |
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165 | (1) |
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5.4.4 Spin glasses and optimization by simulated annealing |
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166 | (5) |
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5.4.5 Ageing in spin glasses and related systems |
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171 | (1) |
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5.4.6 Vector spin glasses: developments and surprises |
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172 | (1) |
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5.5 Models with mixed degrees of freedom: Si/Ge alloys, a case study |
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173 | (1) |
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5.6 Sampling the free energy and entropy |
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174 | (4) |
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5.6.1 Thermodynamic integration |
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174 | (2) |
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5.6.2 Groundstate free energy determination |
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176 | (1) |
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5.6.3 Estimation of intensive variables: the chemical potential |
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177 | (1) |
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5.6.4 Lee-Kosterlitz method |
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177 | (1) |
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5.6.5 Free energy from finite size dependence at TT |
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178 | (1) |
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178 | (15) |
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5.7.1 Inhomogeneous systems: surfaces, interfaces, etc. |
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178 | (6) |
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5.7.2 Other Monte Carlo schemes |
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184 | (2) |
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5.7.3 Inverse and reverse Monte Carlo methods |
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186 | (1) |
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5.7.4 Finite size effects: a review and summary |
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187 | (1) |
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5.7.5 More about error estimation |
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188 | (2) |
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5.7.6 Random number generators revisited |
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190 | (3) |
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5.8 Summary and perspective |
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193 | (1) |
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193 | (4) |
6 Off-lattice models |
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197 | (60) |
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197 | (28) |
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6.1.1 NIT ensemble and the virial theorem |
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197 | (3) |
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200 | (4) |
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6.1.3 Grand canonical ensemble |
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204 | (4) |
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6.1.4 Near critical coexistence: a case study |
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208 | (2) |
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6.1.5 Subsystems: a case study |
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210 | (5) |
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215 | (3) |
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6.1.7 Widom particle insertion method and variants |
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218 | (2) |
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6.1.8 Monte Carlo Phase Switch |
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220 | (4) |
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6.1.9 Cluster algorithm for fluids |
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224 | (1) |
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6.2 'Short range' interactions |
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225 | (1) |
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225 | (1) |
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6.2.2 Verlet tables and cell structure |
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225 | (1) |
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6.2.3 Minimum image convention |
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226 | (1) |
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6.2.4 Mixed degrees of freedom reconsidered |
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226 | (1) |
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6.3 Treatment of long range forces |
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226 | (3) |
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6.3.1 Reaction field method |
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226 | (1) |
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227 | (1) |
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6.3.3 Fast multipole method |
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228 | (1) |
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229 | (2) |
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229 | (1) |
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6.4.2 Periodic substrate potentials |
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229 | (2) |
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231 | (3) |
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6.5.1 Application of the Liu-Luijten algorithm to a binary fluid mixture |
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233 | (1) |
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6.6 Polymers: an introduction |
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234 | (16) |
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6.6.1 Length scales and models |
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234 | (7) |
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6.6.2 Asymmetric polymer mixtures: a case study |
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241 | (4) |
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6.6.3 Applications: dynamics of polymer melts; thin adsorbed polymeric films |
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245 | (3) |
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6.6.4 Polymer melts: speeding up bond fluctuation model simulations |
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248 | (2) |
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6.7 Configurational bias and 'smart Monte Carlo' |
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250 | (3) |
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253 | (1) |
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253 | (4) |
7 Reweighting methods |
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257 | (28) |
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257 | (3) |
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7.1.1 Distribution functions |
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257 | (1) |
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257 | (3) |
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7.2 Single histogram method: the Ising model as a case study |
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260 | (7) |
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7.3 Multihistogram method |
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267 | (1) |
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7.4 Broad histogram method |
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268 | (1) |
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7.5 Transition matrix Monte Carlo |
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268 | (1) |
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7.6 Multicanonical sampling |
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269 | (5) |
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7.6.1 The multicanonical approach and its relationship to canonical sampling |
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269 | (1) |
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7.6.2 Near first order transitions |
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270 | (2) |
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7.6.3 Groundstates in complicated energy landscapes |
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272 | (1) |
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7.6.4 Interface free energy estimation |
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273 | (1) |
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7.7 A case study: the Casimir effect in critical systems |
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274 | (2) |
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7.8 'Wang-Landau sampling' |
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276 | (5) |
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276 | (3) |
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7.8.2 Applications to models with continuous variables |
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279 | (1) |
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7.8.3 Case studies with two-dimensional Wang-Landau sampling |
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279 | (1) |
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7.8.4 Back to numerical integration |
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279 | (2) |
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7.9 A case study: evaporation/condensation transition of droplets |
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281 | (1) |
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282 | (3) |
8 Quantum Monte Carlo methods |
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285 | (39) |
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285 | (2) |
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8.2 Feynman path integral formulation |
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287 | (10) |
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8.2.1 Off-lattice problems: low-temperature properties of crystals |
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287 | (6) |
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8.2.2 Bose statistics and superfluidity |
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293 | (1) |
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8.2.3 Path integral formulation for rotational degrees of freedom |
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294 | (3) |
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297 | (19) |
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8.3.1 The Ising model in a transverse field |
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297 | (1) |
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8.3.2 Anisotropic Heisenberg chain |
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298 | (4) |
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8.3.3 Fermions on a lattice |
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302 | (2) |
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8.3.4 An intermezzo: the minus sign problem |
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304 | (2) |
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8.3.5 Spinless fermions revisited |
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306 | (4) |
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8.3.6 Cluster methods for quantum lattice models |
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310 | (1) |
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8.3.7 Continuous time simulations |
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310 | (1) |
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8.3.8 Decoupled cell method |
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311 | (1) |
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312 | (1) |
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8.3.10 Wang-Landau sampling for quantum models |
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313 | (1) |
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8.3.11 Fermion determinants |
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314 | (2) |
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8.4 Monte Carlo methods for the study of groundstate properties |
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316 | (4) |
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8.4.1 Variational Monte Carlo (VMC) |
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316 | (2) |
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8.4.2 Green's function Monte Carlo methods (GFMC) |
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318 | (2) |
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320 | (1) |
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321 | (3) |
9 Monte Carlo renormalization group methods |
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324 | (14) |
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9.1 Introduction to renormalization group theory |
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324 | (4) |
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9.2 Real space renormalization group |
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328 | (1) |
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9.3 Monte Carlo renormalization group |
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329 | (7) |
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9.3.1 Large cell renormalization |
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329 | (2) |
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9.3.2 Ma's method: finding critical exponents and the fixed point Hamiltonian |
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331 | (1) |
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332 | (2) |
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9.3.4 Location of phase boundaries |
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334 | (1) |
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9.3.5 Dynamic problems: matching time-dependent correlation functions |
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335 | (1) |
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9.3.6 Inverse Monte Carlo renormalization group transformations |
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336 | (1) |
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336 | (2) |
10 Non-equilibrium and irreversible processes |
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338 | (27) |
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10.1 Introduction and perspective |
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338 | (1) |
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10.2 Driven diffusive systems (driven lattice gases) |
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338 | (3) |
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341 | (3) |
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344 | (3) |
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347 | (2) |
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347 | (1) |
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347 | (2) |
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10.6 Growth of structures and patterns |
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349 | (4) |
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10.6.1 Eden model of cluster growth |
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349 | (1) |
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10.6.2 Diffusion limited aggregation |
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349 | (3) |
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10.6.3 Cluster-cluster aggregation |
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352 | (1) |
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352 | (1) |
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10.7 Models for film growth |
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353 | (5) |
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353 | (1) |
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10.7.2 Ballistic deposition |
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354 | (1) |
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355 | (1) |
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10.7.4 Kinetic Monte Carlo and MBE growth |
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356 | (2) |
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10.8 Transition path sampling |
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358 | (1) |
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10.9 Forced polymer pore translocation: a case study |
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359 | (3) |
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10.10 Outlook: variations on a theme |
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362 | (1) |
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362 | (3) |
11 Lattice gauge models: a brief introduction |
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365 | (14) |
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11.1 Introduction: gauge invariance and lattice gauge theory |
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365 | (2) |
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11.2 Some technical matters |
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367 | (1) |
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11.3 Results for Z(N) lattice gauge models |
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367 | (1) |
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11.4 Compact U(1) gauge theory |
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368 | (1) |
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11.5 SU(2) lattice gauge theory |
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369 | (1) |
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11.6 Introduction: quantum chromodynamics (QCD) and phase transitions of nuclear matter |
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370 | (2) |
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11.7 The deconfinement transition of QCD |
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372 | (3) |
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11.8 Towards quantitative predictions |
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375 | (2) |
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377 | (2) |
12 A brief review of other methods of computer simulation |
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379 | (22) |
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379 | (1) |
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379 | (9) |
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12.2.1 Integration methods (microcanonical ensemble) |
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379 | (4) |
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12.2.2 Other ensembles (constant temperature, constant pressure, etc.) |
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383 | (3) |
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12.2.3 Non-equilibrium molecular dynamics |
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386 | (1) |
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12.2.4 Hybrid methods (MD + MC) |
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386 | (1) |
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12.2.5 Ab initio molecular dynamics |
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387 | (1) |
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12.2.6 Hyperdynamics and metadynamics |
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388 | (1) |
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12.3 Quasi-classical spin dynamics |
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388 | (4) |
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12.4 Langevin equations and variations (cell dynamics) |
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392 | (1) |
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393 | (1) |
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12.6 Dissipative particle dynamics (DPD) |
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393 | (2) |
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12.7 Lattice gas cellular automata |
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395 | (1) |
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12.8 Lattice Boltzmann equation |
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395 | (1) |
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12.9 Multiscale simulation |
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396 | (2) |
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398 | (3) |
13 Monte Carlo simulations at the periphery of physics and beyond |
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401 | (18) |
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401 | (1) |
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401 | (1) |
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402 | (1) |
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403 | (2) |
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13.5 'Biologically inspired' physics |
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405 | (4) |
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13.5.1 Commentary and perspective |
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405 | (1) |
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406 | (1) |
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407 | (2) |
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409 | (1) |
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13.7 Mathematics/statistics |
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410 | (1) |
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410 | (1) |
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411 | (1) |
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13.10 'Traffic' simulations |
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412 | (1) |
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413 | (1) |
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13.12 Networks: what connections really matter? |
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414 | (1) |
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415 | (1) |
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416 | (3) |
14 Monte Carlo studies of biological molecules |
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419 | (11) |
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419 | (1) |
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420 | (6) |
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420 | (1) |
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14.2.2 How to best simulate proteins: Monte Carlo or molecular dynamics |
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421 | (1) |
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14.2.3 Generalized ensemble methods |
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421 | (2) |
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14.2.4 Globular proteins: a case study |
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423 | (1) |
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14.2.5 Simulations of membrane proteins |
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424 | (2) |
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14.3 Monte Carlo simulations of carbohydrates |
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426 | (1) |
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14.4 Determining macromolecular structures |
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427 | (1) |
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428 | (1) |
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428 | (2) |
15 Outlook |
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430 | (3) |
Appendix: listing of programs mentioned in the text |
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433 | (32) |
Index |
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465 | |