<p> CONTENTS Notation xi</p><p>Preface xiii</p><p>Chapter 1 Background, Mechanics and Statistical Mechanics 1</p><p>1.1 Background 1</p><p>1.2 The Mechanical Description of a System of Particles 5</p><p>1.2.1 Phase Space and Equations of Motion 10</p><p>1.2.2 In Equilibrium 10</p><p>1.2.3 In Non-equilibrium 12</p><p>1.2.4 Newton's Equations in Operator Form 14</p><p>1.2.5 The Liouville Equation 15</p><p>1.2.6 Liouville Equation in Equilibrium 15</p><p>1.2.7 Expressions for Equilibrium Thermodynamic 16</p><p> and Linear Transport Properties</p><p>1.2.8 Liouville Equation in Non-equilibrium 16</p><p>1.2.9 Nonequilibrium Distribution Function 17</p><p> and Correlation Functions </p><p>1.2.10 Other Approaches to Non-equilibrium 20</p><p>1.2.11 Projection Operators 20</p><p>1.3 Summary 22</p><p>Conclusions 23</p><p>References 24</p><p>Chapter 2 The Equation of Motion for a Typical Particle at Equilibrium – 26 The Mori-Zwanzig Approach </p><p>2.1 The Projection Operator 26</p><p>2.2 The Generalised Langevin Equation 28 2.3 The Generalised Langevin Equation in Terms of the Velocity 31</p><p>2.4 Equation of Motion for the Velocity Autocorrelation Function 32</p><p>2.5 The Langevin Equation Derived from the Mori Approach 34</p><p>– The Brownian Limit</p><p>2.6 Generalization to Any Set of Dynamical Variables 35</p><p>2.7 Memory Functions Derivation of Expressions for 38</p><p>Linear Transport Coefficients</p><p>2.8 Correlation Function Expression for the 39</p><p>Coefficient of Newtonian Viscosity</p><p>2.9 Summary 43</p><p>Conclusions 44</p><p>References 46</p><p>Chapter 3 Approximate Methods to Calculate Correlation Functions 48 And Mori-Zwanzig Memory Functions</p><p>3.1 Taylor Series Expansion 48</p><p>3.2 Spectra 50</p><p>3.3 Mori Continued Fraction Method 51</p><p>3.4 Use of Information Theory 53</p><p>3.5 Perturbation Theories 55</p><p>3.6 Mode Coupling Theory 59</p><p>3.7 Macroscopic Hydrodynamic Theory 60</p><p>3.8 Memory Functions Calculated by the MD Method 64</p><p>Conclusions 65</p><p>References 66</p><p>Chapter 4 The Generalised Langevin Equation in Non-equilibrium 70 4.1 Derivation of Generalized Langevin Equation in 71</p><p>Non-Equilibrium</p><p>4.2 Langevin Equation for a Single Brownian Particle 75</p><p> in a Shearing Fluid</p><p>Conclusions 81</p><p>References 82</p><p>Chapter 5 The Langevin Equation and The Brownian Limit 83</p><p>5.1 A Dilute Suspension – One Large Particle in a Background 84</p><p>5.1.1 Exact Equations of Motion for A(t) 88</p><p>5.1.2 Langevin Equation for A(t) 89</p><p>5.1.3 Langevin Equation for Velocity 92</p><p>5.2 Many Body Langevin Equation 96</p><p>5.2.1 Exact Equations of Motion for A(t) 99</p><p>5.2.2 Many-Body Langevin Equation for A(t) 101</p><p>5.2.3 Many-Body Langevin Equation for Velocity 102</p><p>5.2.4 Langevin Equation for the Velocity and 104</p><p> the Form of the Friction Coefficients</p><p>5.3 Generalization to Non-Equilibrium 106</p><p>5.4 The Fokker-Planck Equation and the Diffusive Limit 108</p><p>5.5 Approach to the Brownian Limit and Limitations 110</p><p>5.5.1 A Basic Limitation of the LE and FP Equation 111</p><p>5.5.2 The Friction Coefficient 112</p><p>5.5.3 Self Diffusion Coefficient, Ds 114</p><p>5.5.4 The Intermediate Scattering Function F(q,t) 117</p><p>5.6 Summary 120 Conclusions 121</p><p>References 122</p><p>Chapter 6 Langevin and Generalised Langevin Dynamics 126</p><p>6.1 Extensions of the GLE to Collections of Particles 126</p><p>6.2 Numerical Solution of the Langevin Equation 129</p><p>6.2.1 Gaussian Random Variables 130</p><p>6.2.2 A BD Algorithm to First Order in t 133</p><p>6.2.3 A Second First Order BD Algorithm 135</p><p>6.2.4 A Third First Order BD Algorithm 137</p><p>6.2.5 The BD Algorithm in the Diffusive Limit 139</p><p>6.3 Higher Order BD Schemes for Langevin Equation 140</p><p>6.4 Generalised Langevin Equation 141</p><p>6.4.1 The Method of Berkowitz, Morgan, and McCammon 142</p><p>6.4.2 The Method of Ermak and Buckholtz 143</p><p>6.4.3 The Method of Ciccotti and Rychaert 145</p><p>6.4.4 Other Methods of Solving the GLE 146</p><p>6.5 Systems in an External Field 146</p><p>6.6 Boundary Conditions in Simulations 148</p><p>6.6.1 PBC in Equilibrium 148</p><p>6.6.2 PBC in Shear Field 150</p><p>6.6.3 PBC in Elongational Flow 151</p><p>Conclusions 151</p><p>References 153</p><p>Chapter 7 Brownian Dynamics 156</p><p>7.1 Fundamentals 156 7.2. Calculation of Hydrodynamic Interactions 158</p><p>7.3 Alternative Approaches to Treat Hydrodynamic Interactions 161</p><p>7.3.1 The Lattice Boltzmann (LB) Approach 162</p><p>7.3.2 Dissipative Particle Dynamics (DPD) 163</p><p>7.4 Brownian Dynamics Algorithms 163</p><p>7.4.1 The Algorithm of Ermak and McCammon 163</p><p>7.4.2 Approximate BD Schemes 167</p><p>7.4.2.1 Algorithms Neglecting Brownian Motion 167</p><p>7.4.3.2 Use of Effective Two Body Tensors 168</p><p>7.4.3.3 "Mean- Field" Approaches 169</p><p>7.4.3.4 Hydrodynamically Dilute 171</p><p> or Free-Draining Regime</p><p>7.5 Brownian Dynamics in a Shear Field 173</p><p>7.6 Limitations of the BD Method 176</p><p>7.7 Alternatives to BD Simulations 177</p><p>7.7.1 Lattice Boltzmann (LB) Approach 177</p><p>7.7.2 Dissipative Particle Dynamics (DPD) 179</p><p>Conclusions 182</p><p>References 184</p><p>Chapter 8 Polymer Dynamics 190</p><p>8.1 Toxvaerd Approach 193</p><p>8.2 Direct Use of Brownian Dynamics 194</p><p>8.3 Rigid Systems 199</p><p>Conclusions 203</p><p>References 204 Chapter 9 Theories Based on Distribution Functions, Master Equations 244 And Stochastic Equations</p><p>9.1 Fokker-Planck Equation 209</p><p>9.2 The Diffusive Limit and the Smoluchowski Equation 210</p><p>9.2.1 Solution of the n-Body Smoluchowski Equation 213</p><p>9.2.2 Position-only Langevin Equation 213</p><p>9.3 Quantum Monte Carlo Method 215</p><p>9.4 Master Equations 224</p><p>9.4.1 The Identification of Elementary Processes 227</p><p>9.4.2 Kinetic MC and Master Equations 230</p><p>9.4.3 KMC procedure with continuum solids 232</p><p>Conclusions 235</p><p>References 238</p><p>Chapter 10 An Overview 245</p><p>Appendices 250</p><p>Appendix A Expressions for Equilibrium Properties, Transport Coefficients 250</p><p>And Scattering Functions</p><p>A.1 Equilibrium Properties 250</p><p>A.2 Expressions for Linear Transport Coefficients 251</p><p>References 251</p><p>A.3 Scattering Functions 253</p><p>A.3.1 Static Structure 253</p><p>A.3.2 Dynamic Scattering 253</p><p>References 255</p><p>Appendix B Some Basic Results about Operators 257</p><p>Appendix C Proofs Required for the GLE for a Selected Particle 260 Appendix D The Langevin Equation from the Mori-Zwanzig Approach 263</p><p>Appendix E The Friction Coefficient and the Friction Factor 265</p><p>Appendix F Mori Coefficients for a Two Component System 267</p><p>F.1 Basics 267</p><p>F.2 Short Time Expansions 267</p><p>F.3 Relative Initial Behaviour of c(t) 268</p><p>Appendix G Time Reversal Symmetry of Non-equilibrium 269 Correlation Functions</p><p>References 271</p><p>Appendix H Some Proofs Needed for the Albers, Deutch and 272 Oppenheim Treatment</p><p>Appendix I Some Proofs Needed for the Deutch and Oppenheim Treatment 275 Appendix J The Calculation of the Bulk Properties of 276 Colloids and Polymers</p><p>J.1 Equilibrium Properties 276</p><p>J.2. Static Structure 276</p><p>J.3. Time Correlation Functions 276</p><p>J.3.1 Self Diffusion 277</p><p>J.3.2 Time Dependent Scattering 277</p><p>J.3.3 Bulk Stress 277</p><p>J.3.4 Zero Time (High Frequency) Results 278</p><p> in the Diffusive Limit</p><p>References 280</p><p>Appendix K Monte Carlo Methods 282</p><p>K.1 Metropolis Monte Carlo Technique 282</p><p>K.2 An MC Routine 285 References 288</p><p>Appendix L The Generation of Random Numbers 289</p><p>L.1 Generation of Random Deviates 289</p><p>References 290</p><p>Appendix M Hydrodynamic Interaction Tensors 291</p><p>M.1 The Oseen Tensor for Two Bodies 291</p><p>M.2 The Rotne-Prager Tensor for Two Bodies 291</p><p>M.3 The Series Result of Jones and Burfield for Two Bodies 291</p><p>M.4 Mazur and van Saarloos Results for Three Bodies 292</p><p>M.5 Results of Lubrication Theory 292</p><p>M.6 The Rotne-Prager Tensor in Periodic Boundary Conditions 292</p><p>References 293</p><p>Appendix N Calculation of Hydrodynamic Interaction Tensors 294</p><p>References 297</p><p>Appendix O Some FORTRAN Programs </p>