The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems Book

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