The Langevin and generalised Langevin approach to the dynamics of atomic, polymeric and colloidal systems / Ian Snook.

Format:

Book

Author/Creator:

Snook, Ian.

Contributor:

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Langevin equations.

Brownian movements.

Random dynamical systems.

Physics.

Physical Description:

xvi, 303 pages : illustrations (some color) ; 25 cm

Edition:

First edition.

Place of Publication:

Amsterdam ; Boston : Elsevier, 2007.

Summary:

The dynamics of atomic systems is frequently studied by the numerical solution of Newton's equation of motion, which is termed the method of Molecular Dynamics. However, many atomic processes are too slow to be treated in this manner particularly for process involving surfaces, polymers and colloidal systems. Thus, methods have been developed to overcome this which are based on averaging the atomic equations of motion over some of the degrees of freedom of the system, typically those which occur on the fastest time scales. This coarse-graining results in differential equations called Generalised Langevin equations which contain stochastic (random) terms which means this averaging has lead to a loss of information but these equations are able to treat problems which cannot be easily tackled by direct atomic simulations.

A Variety of such coarse-grained methods have been developed and it is shown that the dynamical equations of the Generalised Langevin method may derived from the microscopic Liouville equation by use of projection operators. Such Generalised Langevin equations are developed for equilibrium systems, non-equilibrium situations and for the limiting case of large particles suspended in an atomic fluid.

Detailed derivations are then given of algorithms to solve these dynamical equations numerical which are termed Generalised Brownian Dynamics and Brownian Dynamics Methods. Examples of the application of these numerical methods to typical atomic, polymeric and colloidal systems are discussed in order to demonstrate the type of information which may be obtained by their use.

In a later chapter alternative methods are discussed which are based on distribution functions which results in the Fokker-Planck equation and, in the diffusive limit, the Smoluchowski or Many-body Diffusion Equation. It is shown that these equations may be used as an alternative way of deriving numerical schemes and for developing schemes to numerically solve to Stationary State Schrodinger equation.

Finally, Methods based on Master Equations are outlined which, instead of starting from atomic equations, are based on the postulate that processes may be treated as inherently stochastic. The techniques needed to implement practical schemes based on these Master Equations are outlined including Transition State Theory, Accelerated Dynamics Methods and the Kinetic Monte Carlo Method.

Extensive appendices are also included which cover many of the details of the derivations outlined in the main text and cover many related topics for example operator techniques, the Metropolis Monte Carlo method and the relationship between dynamics and bulk properties. Also included are some simple computer programs which enabled the calculation of many quantities discussed in the text.

Contents:

1 Background, Mechanics and Statistical Mechanics 1

1.2 The Mechanical Description of a System of Particles 3

1.2.1 Phase space and equations of motion 7

1.2.2 In equilibrium 7

1.2.3 In a non-isolated system 9

1.2.4 Newton's equations in operator form 10

1.2.5 The Liouville equation 11

1.2.6 Liouville equation in an isolated system 11

1.2.7 Expressions for equilibrium thermodynamic and linear transport properties 11

1.2.8 Liouville equation in a non-isolated system 12

1.2.9 Non-equilibrium distribution function and correlation functions 13

1.2.10 Other approaches to non-equilibrium 15

1.2.11 Projection operators 15

2 The Equation of Motion for a Typical Particle at Equilibrium: The Mori-Zwanzig Approach 21

2.1 The Projection Operator 21

2.2 The Generalised Langevin Equation 23

2.3 The Generalised Langevin Equation in Terms of the Velocity 26

2.4 Equation of Motion for the Velocity Autocorrelation Function 28

2.5 The Langevin Equation Derived from the Mori Approach: The Brownian Limit 29

2.6 Generalisation to any Set of Dynamical Variables 30

2.7 Memory Functions Derivation of Expressions for Linear Transport Coefficients 33

2.8 Correlation Function Expression for the Coefficient of Newtonian Viscosity 34

3 Approximate Methods to Calculate Correlation Functions and Mori-Zwanzig Memory Functions 41

3.1 Taylor Series Expansion 41

3.2 Spectra 43

3.3 Mori's Continued Fraction Method 44

3.4 Use of Information Theory 46

3.5 Perturbation Theories 48

3.6 Mode Coupling Theory 51

3.7 Macroscopic Hydrodynamic Theory 52

3.8 Memory Functions Calculated by the Molecular-Dynamics Method 56

4 The Generalised Langevin Equation in Non-Equilibrium 61

4.1 Derivation of Generalised Langevin Equation in Non-Equilibrium 62

4.2 Langevin Equation for a Single Brownian Particle in a Shearing Fluid 66

5 The Langevin Equation and the Brownian Limit 71

5.1 A Dilute Suspension - One Large Particle in a Background 72

5.1.1 Exact equations of motion for A(t) 75

5.1.2 Langevin equation for A(t) 77

5.1.3 Langevin equation for velocity 80

5.2 Many-body Langevin Equation 83

5.2.1 Exact equations of motion for A(t) 87

5.2.2 Many-body Langevin equation for A(t) 89

5.2.3 Many-body Langevin equation for velocity 90

5.2.4 Langevin equation for the velocity and the form of the friction coefficients 92

5.3 Generalisation to Non-Equilibrium 94

5.4 The Fokker-Planck Equation and the Diffusive Limit 95

5.5 Approach to the Brownian Limit and Limitations 97

5.5.1 A basic limitation of the LE and FP equations 98

5.5.2 The friction coefficient 98

5.5.3 Self-diffusion coefficient (D[subscript s]) 99

5.5.4 The intermediate scattering function F(q,t) 102

5.5.5 Systems in a shear field 102

6 Langevin and Generalised Langevin Dynamics 107

6.1 Extensions of the GLE to Collections of Particles 107

6.2 Numerical Solution of the Langevin Equation 110

6.2.1 Gaussian random variables 111

6.2.2 A BD algorithm to first-order in [Delta]t 113

6.2.3 A second first-order BD algorithm 116

6.2.4 A third first-order BD algorithm 118

6.2.5 The BD algorithm in the diffusive limit 120

6.3 Higher-Order BD Schemes for the Langevin Equation 120

6.4 Generalised Langevin Equation 121

6.4.1 The method of Berkowitz, Morgan and McCammon 122

6.4.2 The method of Ermak and Buckholz 123

6.4.3 The method of Ciccotti and Ryckaert 125

6.4.4 Other methods of solving the GLE 126

6.5 Systems in an External Field 127

6.6 Boundary Conditions in Simulations 128

6.6.1 PBC in equilibrium 128

6.6.2 PBC in a shear field 129

6.6.3 PBC in elongational flow 129

7 Brownian Dynamics 133

7.2 Calculation of Hydrodynamic Interactions 135

7.3 Alternative Approaches to Treat Hydrodynamic Interactions 137

7.3.1 The lattice Boltzmann approach 138

7.3.2 Dissipative particle dynamics 138

7.4 Brownian Dynamics Algorithms 138

7.4.1 The algorithm of Ermak and McCammon 138

7.4.2 Approximate BD schemes 142

7.5 Brownian Dynamics in a Shear Field 146

7.6 Limitations of the BD Method 148

7.7 Alternatives to BD Simulations 149

7.7.1 Lattice Boltzmann approach 149

7.7.2 Dissipative particle dynamics 150

8 Polymer Dynamics 157

8.1 Toxvaerd Approach 159

8.2 Direct Use of Brownian Dynamics 160

8.3 Rigid Systems 163

9 Theories Based on Distribution Functions, Master Equations and Stochastic Equations 169

9.1 Fokker-Planck Equation 170

9.2 The Diffusive Limit and the Smoluchowski Equation 171

9.2.1 Solution of the n-body Smoluchowski equation 173

9.2.2 Position-only Langevin equation 174

9.3 Quantum Monte Carlo Method 176

9.4 Master Equations 180

9.4.1 The identification of elementary processes 184

9.4.2 Kinetic MC and master equations 186

9.4.3 KMC procedure with continuum solids 187

Appendix A Expressions for Equilibrium Properties, Transport Coefficients and Scattering Functions 201

A.1 Equilibrium Properties 201

A.2 Expressions for Linear Transport Coefficients 202

A.3 Scattering Functions 204

A.3.1 Static structure 204

A.3.2 Dynamic scattering 204

Appendix B Some Basic Results About Operators 209

Appendix C Proofs Required for the GLE for a Selected Particle 213

Appendix D The Langevin Equation from the Mori-Zwanzig Approach 217

Appendix E The Friction Coefficient and Friction Factor 221

Appendix F Mori Coefficients for a Two-Component System 223

F.2 Short Time Expansions 224

F.3 Relative Initial Behaviour of c(t) 224

Appendix G Time-Reversal Symmetry of Non-Equilibrium Correlation Functions 225

Appendix H Some Proofs Needed for the Albers, Deutch and Oppenheim Treatment 229

Appendix I A Proof Needed for the Deutch and Oppenheim Treatment 233

Appendix J The Calculation of the Bulk Properties of Colloids and Polymers 235

J.1 Equilibrium Properties 235

J.2 Static Structure 235

J.3 Time Correlation Functions 236

J.3.1 Self-diffusion 236

J.3.2 Time-dependent scattering 236

J.3.3 Bulk stress 237

J.3.4 Zero time (high frequency) results in the diffusive limit 237

Appendix K Monte Carlo Methods 241

K.1 Metropolis Monte Carlo Technique 241

K.2 An MC Routine 243

Appendix L The Generation of Random Numbers 249

L.1 Generation of Random Deviates for BD Simulations 249

Appendix M Hydrodynamic Interaction Tensors 251

M.1 The Oseen Tensor for Two Bodies 251

M.2 The Rotne-Prager Tensor for Two Bodies 251

M.3 The Series Result of Jones and Burfield for Two Bodies 251

M.4 Mazur and Van Saarloos Results for Three Bodies 252

M.5 Results of Lubrication Theory 252

M.6 The Rotne-Prager Tensor in Periodic Boundary Conditions 253

Appendix N Calculation of Hydrodynamic Interaction Tensors 255

Appendix O Some Fortran Programs 261.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0444521291

9780444521293

OCLC:

70176954

Online:

Publisher description