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