Preface |
|
xiii | |
Notation |
|
xv | |
|
Background, Mechanics and Statistical Mechanics |
|
|
1 | (20) |
|
|
1 | (2) |
|
The Mechanical Description of a System of Particles |
|
|
3 | (13) |
|
Phase space and equations of motion |
|
|
7 | (1) |
|
|
7 | (2) |
|
|
9 | (1) |
|
Newton's equations in operator form |
|
|
10 | (1) |
|
|
11 | (1) |
|
Liouville equation in an isolated system |
|
|
11 | (1) |
|
Expressions for equilibrium thermodynamic and linear transport properties |
|
|
11 | (1) |
|
Liouville equation in a non-isolated system |
|
|
12 | (1) |
|
Non-equilibrium distribution function and correlation functions |
|
|
13 | (2) |
|
Other approaches to non-equilibrium |
|
|
15 | (1) |
|
|
15 | (1) |
|
|
16 | (2) |
|
|
18 | (3) |
|
|
18 | (3) |
|
The Equation of Motion for a Typical Particle at Equilibrium: The Mori-Zwanzig Approach |
|
|
21 | (20) |
|
|
21 | (2) |
|
The Generalised Langevin Equation |
|
|
23 | (3) |
|
The Generalised Langevin Equation in Terms of the Velocity |
|
|
26 | (2) |
|
Equation of Motion for the Velocity Autocorrelation Function |
|
|
28 | (1) |
|
The Langevin Equation Derived from the Mori Approach: The Brownian Limit |
|
|
29 | (1) |
|
Generalisation to any Set of Dynamical Variables |
|
|
30 | (3) |
|
Memory Functions Derivation of Expressions for Linear Transport Coefficients |
|
|
33 | (1) |
|
Correlation Function Expression for the Coefficient of Newtonian Viscosity |
|
|
34 | (4) |
|
|
38 | (1) |
|
|
39 | (2) |
|
|
39 | (2) |
|
Approximate Methods to Calculate Correlation Functions and Mori-Zwanzig Memory Functions |
|
|
41 | (20) |
|
|
41 | (2) |
|
|
43 | (1) |
|
Mori's Continued Fraction Method |
|
|
44 | (2) |
|
Use of Information Theory |
|
|
46 | (2) |
|
|
48 | (3) |
|
|
51 | (1) |
|
Macroscopic Hydrodynamic Theory |
|
|
52 | (4) |
|
Memory Functions Calculated by the Molecular-Dynamics Method |
|
|
56 | (1) |
|
|
57 | (4) |
|
|
57 | (4) |
|
The Generalised Langevin Equation in Non-Equilibrium |
|
|
61 | (10) |
|
Derivation of Generalised Langevin Equation in Non-Equilibrium |
|
|
62 | (4) |
|
Langevin Equation for a Single Brownian Particle in a Shearing Fluid |
|
|
66 | (3) |
|
|
69 | (2) |
|
|
69 | (2) |
|
The Langevin Equation and the Brownian Limit |
|
|
71 | (36) |
|
A Dilute Suspension -- One Large Particle in a Background |
|
|
72 | (11) |
|
Exact equations of motion for A (t) |
|
|
75 | (2) |
|
Langevin equation for A(t) |
|
|
77 | (3) |
|
Langevin equation for velocity |
|
|
80 | (3) |
|
Many-body Langevin Equation |
|
|
83 | (11) |
|
Exact equations of motion for A(t) |
|
|
87 | (2) |
|
Many-body Langevin equation for A(t) |
|
|
89 | (1) |
|
Many-body Langevin equation for velocity |
|
|
90 | (2) |
|
Langevin equation for the velocity and the form of the friction coefficients |
|
|
92 | (2) |
|
Generalisation to Non-Equilibrium |
|
|
94 | (1) |
|
The Fokker-Planck Equation and the Diffusive Limit |
|
|
95 | (2) |
|
Approach to the Brownian Limit and Limitations |
|
|
97 | (7) |
|
A basic limitation of the LE and FP equations |
|
|
98 | (1) |
|
|
98 | (1) |
|
Self-diffusion coefficient (Ds) |
|
|
99 | (3) |
|
The intermediate scattering function F(q,t) |
|
|
102 | (1) |
|
|
102 | (2) |
|
|
104 | (1) |
|
|
104 | (3) |
|
|
105 | (2) |
|
Langevin and Generalised Langevin Dynamics |
|
|
107 | (26) |
|
Extensions of the GLE to Collections of Particles |
|
|
107 | (3) |
|
Numerical Solution of the Langevin Equation |
|
|
110 | (10) |
|
Gaussian random variables |
|
|
111 | (2) |
|
A BD algorithm to first-order in Δt |
|
|
113 | (3) |
|
A second first-order BD algorithm |
|
|
116 | (2) |
|
A third first-order BD algorithm |
|
|
118 | (2) |
|
The BD algorithm in the diffusive limit |
|
|
120 | (1) |
|
Higher-Order BD Schemes for the Langevin Equation |
|
|
120 | (1) |
|
Generalised Langevin Equation |
|
|
121 | (6) |
|
The method of Berkowitz, Morgan and McCammon |
|
|
122 | (1) |
|
The method of Ermak and Buckholz |
|
|
123 | (2) |
|
The method of Ciccotti and Ryckaert |
|
|
125 | (1) |
|
Other methods of solving the GLE |
|
|
126 | (1) |
|
Systems in an External Field |
|
|
127 | (1) |
|
Boundary Conditions in Simulations |
|
|
128 | (3) |
|
|
128 | (1) |
|
|
129 | (1) |
|
|
129 | (2) |
|
|
131 | (2) |
|
|
131 | (2) |
|
|
133 | (24) |
|
|
133 | (2) |
|
Calculation of Hydrodynamic Interactions |
|
|
135 | (2) |
|
Alternative Approaches to Treat Hydrodynamic Interactions |
|
|
137 | (1) |
|
The lattice Boltzmann approach |
|
|
138 | (1) |
|
Dissipative particle dynamics |
|
|
138 | (1) |
|
Brownian Dynamics Algorithms |
|
|
138 | (8) |
|
The algorithm of Ermak and McCammon |
|
|
138 | (4) |
|
|
142 | (4) |
|
Brownian Dynamics in a Shear Field |
|
|
146 | (2) |
|
Limitations of the BD Method |
|
|
148 | (1) |
|
Alternatives to BD Simulations |
|
|
149 | (3) |
|
Lattice Boltzmann approach |
|
|
149 | (1) |
|
Dissipative particle dynamics |
|
|
150 | (2) |
|
|
152 | (5) |
|
|
153 | (4) |
|
|
157 | (12) |
|
|
159 | (1) |
|
Direct Use of Brownian Dynamics |
|
|
160 | (3) |
|
|
163 | (3) |
|
|
166 | (3) |
|
|
166 | (3) |
|
Theories Based on Distribution Functions, Master Equations and Stochastic Equations |
|
|
169 | (28) |
|
|
170 | (1) |
|
The Diffusive Limit and the Smoluchowski Equation |
|
|
171 | (5) |
|
Solution of the η-body Smoluchowski equation |
|
|
173 | (1) |
|
Position-only Langevin equation |
|
|
174 | (2) |
|
Quantum Monte Carlo Method |
|
|
176 | (4) |
|
|
180 | (9) |
|
The identification of elementary processes |
|
|
184 | (2) |
|
Kinetic MC and master equations |
|
|
186 | (1) |
|
KMC procedure with continuum solids |
|
|
187 | (2) |
|
|
189 | (8) |
|
|
191 | (6) |
|
|
197 | (4) |
|
Appendix A: Expressions for Equilibrium Properties, Transport Coefficients and Scattering Functions |
|
|
201 | (8) |
|
|
201 | (1) |
|
Expressions for Linear Transport Coefficients |
|
|
202 | (2) |
|
|
204 | (5) |
|
|
204 | (1) |
|
|
204 | (2) |
|
|
206 | (3) |
|
Appendix B: Some Basic Results About Operators |
|
|
209 | (4) |
|
Appendix C: Proofs Required for the GLE for a Selected Particle |
|
|
213 | (4) |
|
Appendix D: The Langevin Equation from the Mori-Zwanzig Approach |
|
|
217 | (4) |
|
Appendix E: The Friction Coefficient and Friction Factor |
|
|
221 | (2) |
|
Appendix F: Mori Coefficients for a Two-Component System |
|
|
223 | (2) |
|
|
223 | (1) |
|
|
224 | (1) |
|
Relative Initial Behaviour of c(t) |
|
|
224 | (1) |
|
Appendix G: Time-Reversal Symmetry of Non-Equilibrium Correlation Functions |
|
|
225 | (4) |
|
|
227 | (2) |
|
Appendix H: Some Proofs Needed for the Albers, Deutch and Oppenheim Treatment |
|
|
229 | (4) |
|
Appendix I: A Proof Needed for the Deutch and Oppenheim Treatment |
|
|
233 | (2) |
|
Appendix J: The Calculation of the Bulk Properties of Colloids and Polymers |
|
|
235 | (6) |
|
|
235 | (1) |
|
|
235 | (1) |
|
Time Correlation Functions |
|
|
236 | (5) |
|
|
236 | (1) |
|
Time-dependent scattering |
|
|
236 | (1) |
|
|
237 | (1) |
|
Zero time (high frequency) results in the diffusive limit |
|
|
237 | (2) |
|
|
239 | (2) |
|
Appendix K: Monte Carlo Methods |
|
|
241 | (8) |
|
Metropolis Monte Carlo Technique |
|
|
241 | (2) |
|
|
243 | (6) |
|
|
248 | (1) |
|
Appendix L: The Generation of Random Numbers |
|
|
249 | (2) |
|
Generation of Random Deviates for BD Simulations |
|
|
249 | (2) |
|
|
250 | (1) |
|
Appendix M: Hydrodynamic Interaction Tensors |
|
|
251 | (4) |
|
The Oseen Tensor for Two Bodies |
|
|
251 | (1) |
|
The Rotne-Prager Tensor for Two Bodies |
|
|
251 | (1) |
|
The Series Result of Jones and Burfield for Two Bodies |
|
|
251 | (1) |
|
Mazur and Van Saarloos Results for Three Bodies |
|
|
252 | (1) |
|
Results of Lubrication Theory |
|
|
252 | (1) |
|
The Rotne-Prager Tensor in Periodic Boundary Conditions |
|
|
253 | (2) |
|
|
253 | (2) |
|
Appendix N: Calculation of Hydrodynamic Interaction Tensors |
|
|
255 | (6) |
|
|
259 | (2) |
|
Appendix O: Some Fortran Programs |
|
|
261 | (40) |
Index |
|
301 | |