Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems (eBook)

Ian Snook (Autor)

eBook Download: EPUB

2006 | 1. Auflage
320 Seiten
Elsevier Science (Verlag)
978-0-08-046792-4 (ISBN)

Infos

The Langevin and Generalised Langevin Approach To The Dynamics Of Atomic, Polymeric And Colloidal Systems is concerned with the description of aspects of the theory and use of so-called random processes to describe the properties of atomic, polymeric and colloidal systems in terms of the dynamics of the particles in the system. It provides derivations of the basic equations, the development of numerical schemes to solve them on computers and gives illustrations of application to typical systems.
Extensive appendices are given to enable the reader to carry out computations to illustrate many of the points made in the main body of the book.

* Starts from fundamental equations
* Gives up-to-date illustration of the application of these techniques to typical systems of interest
* Contains extensive appendices including derivations, equations to be used in practice and elementary computer codes

Inhalt

Cover 1
The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems 4
Copyright page 5
Contents 8
Preface 14
Notation 16
A. Potential Energy Functions 16
B. Symbols Used 17
C. Operations 17
Chapter 1. Background, Mechanics and Statistical Mechanics 18
1.1 Background 18
1.2 The Mechanical Description of a System of Particles 20
1.3 Summary 33
1.4. Conclusions 35
References 35
Chapter 2. The Equation of Motion for a Typical Particle at Equilibrium:The Mori–Zwanzig Approach 38
2.1 The Projection Operator 38
2.2 The Generalised Langevin Equation 40
2.3 The Generalised Langevin Equation in Terms of the Velocity 43
2.4 Equation of Motion for the Velocity Autocorrelation Function 45
2.5 The Langevin Equation Derived from the Mori Approach: The Brownian Limit 46
2.6 Generalisation to any Set of Dynamical Variables 47
2.7 Memory Functions Derivation of Expressions for Linear Transport Coefficients 50
2.8 Correlation Function Expression for the Coefficient of Newtonian Viscosity 51
2.9 Summary 55
2.10 Conclusions 56
References 56
Chapter 3. Approximate Methods to Calculate Correlation Functions and Mori-Zwanzig Memory Functions 58
3.1 Taylor Series Expansion 58
3.2 Spectra 60
3.3 Mori.s Continued Fraction Method 61
3.4 Use of Information Theory 63
3.5 Perturbation Theories 65
3.6 Mode Coupling Theory 68
3.7 Macroscopic Hydrodynamic Theory 69
3.8 Memory Functions Calculated by the Molecular-Dynamics Method 73
3.9 Conclusions 74
References 74
Chapter 4. The Generalised Langevin Equation in Non-Equilibrium 78
4.1 Derivation of Generalised Langevin Equation in Non-Equilibrium 79
4.2 Langevin Equation for a Single Brownian Particle in a Shearing Fluid 83
4.3 Conclusions 86
References 86
Chapter 5. The Langevin Equation and the Brownian Limit 88
5.1 A Dilute Suspension – One Large Particle in a Background 89
5.2 Many-Body Langevin Equation 100
5.3 Generalisation to Non-Equilibrium 111
5.4 The Fokker–Planck Equation and the Diffusive Limit 112
5.5 Approach to the Brownian Limit and Limitations 114
5.6 Summary 121
5.7 Conclusions 121
References 122
Chapter 6. Langevin and Generalised Langevin Dynamics 124
6.1 Extensions of the GLE to Collections of Particles 124
6.2 Numerical Solution of the Langevin Equation 127
6.3 Higher-Order BD Schemes for the Langevin Equation 137
6.4 Generalised Langevin Equation 138
6.5 Systems in an External Field 144
6.6 Boundary Conditions in Simulations 145
6.7 Conclusions 148
References 148
Chapter 7. Brownian Dynamics 150
7.1 Fundamentals 150
7.2 Calculation of Hydrodynamic Interactions 152
7.3 Alternative Approaches to Treat Hydrodynamic Interactions 154
7.4 Brownian Dynamics Algorithms 155
7.5 Brownian Dynamics in a Shear Field 163
7.6 Limitations of the BD Method 165
7.7 Alternatives to BD Simulations 166
7.8 Conclusions 169
References 170
Chapter 8. Polymer Dynamics 174
8.1 Toxvaerd Approach 176
8.2 Direct Use of Brownian Dynamics 177
8.3 Rigid Systems 180
8.4 Conclusions 183
References 183
Chapter 9. Theories Based on Distribution Functions, Master Equations and Stochastic Equations 186
9.1 Fokker–Planck Equation 187
9.2 The Diffusive Limit and the Smoluchowski Equation 188
9.3 Quantum Monte Carlo Method 193
9.4 Master Equations 197
9.5 Conclusions 206
References 208
Chapter 10. An Overview 214
Appendix A: Expressions for Equilibrium Properties, Transport Coefficients and Scattering Functions 218
A.1 Equilibrium Properties 218
A.2 Expressions for Linear Transport Coefficients 219
A.3 Scattering Functions 221
References 223
Appendix B: Some Basic Results About Operators 226
Appendix C: Proofs Required for the GLE for a Selected Particle 230
Appendix D: The Langevin Equation from the Mori–Zwanzig Approach 234
Appendix E: The Friction Coefficient and Friction Factor 238
Appendix F: Mori Coefficients for a Two-Component System 240
F.1 Basics 240
F.2 Short Time Expansions 241
F.3 Relative Initial Behaviour of c(t) 241
Appendix G: Time-Reversal Symmetry of Non-Equilibrium Correlation Functions 242
References 244
Appendix H: Some Proofs Needed for the Albers, Deutch and Oppenheim Treatment 246
Appendix I: A Proof Needed for the Deutch and Oppenheim Treatment 250
Appendix J: The Calculation of the Bulk Properties of Colloids and Polymers 252
J.1 Equilibrium Properties 252
J.2 Static Structure 252
J.3 Time Correlation Functions 253
References 256
Appendix K: Monte Carlo Methods 258
K.1 Metropolis Monte Carlo Technique 258
K.2 An MC Routine 260
References 265
Appendix L: The Generation of Random Numbers 266
L.1 Generation of Random Deviates for BD Simulations 266
References 267
Appendix M: Hydrodynamic Interaction Tensors 268
M.1 The Oseen Tensor for Two Bodies 268
M.2 The Rotne–Prager Tensor for Two Bodies 268
M.3 The Series Result of Jones and Burfield for Two Bodies 268
M.4 Mazur and van Saarloos Results for Three Bodies 269
M.5 Results of Lubrication Theory 269
M.6 The Rotne–Prager Tensor in Periodic Boundary Conditions 270
References 270
Appendix N: Calculation of Hydrodynamic Interaction Tensors 272
References 276
Appendix O: Some Fortran Programs 278
Index 318

Textauszug

1

Background, Mechanics and Statistical Mechanics

In order to fully appreciate how to calculate and use generalised Langevin equations(GLEs)it is first necessary to review the mechanics upon which these GLEs are based and the statistical mechanics which is used in order to calculate bulk properties from the information which these equations generate.

1.1 BACKGROUND

The instantaneous mechanical state of system described by classical mechanics requires only the specification of a set of positions and momenta of the particles making up the system and provided that these particles are “heavy enough” this classical mechanical approach will provide an accurate description of the physical state of a many-body system. In practice this applies to systems consisting of most atoms under normal physical conditions, except for hydrogen and helium. Then the common approach to describing the time evolution of this mechanical state of such a many-body system, its dynamics, is by use of a coupled set of differential equations, for example, Newton’s equations of motion, which describes the detailed, individual dynamics of all the particles in the system. In the study of molecular systems this approach has led to the development of the widely used numerical technique called molecular dynamics (MD). This method has provided numerous insights into the behaviour of molecular systems and there is now an extensive literature on the method and its application.

However an alternative approach to describing dynamics is to use equations of motion that describe the dynamics of only some, selected particles moving in the presence of the other particles in the system which are now regarded as a background or bath whose detailed dynamics is not treated. Thus, we select out a typical particle or set of particles in which we are interested and find an equation which describes the dynamics of these chosen particles in the presence of the other particles. The classical example of this is the Langevin equation (LE) developed in a heuristic way by Paul Langevin to describe the Brownian motion of a large particle suspended in a fluid consisting of an enormously large number of lighter particles.

The LE is an equation of motion for the velocity of a single Brownian (B) particle of mass MBsuspended in a “bath” consisting of an enormous number, Nb, of particles of much smaller mass mand is

(1.1)

where ζ is the friction factor, –ζthe drag force due to the bath particles and )the random force due to random thermal motion of the bath particles. Thus, instead of a description of the dynamics of the system by writing down the coupled set of Newton equations for the total system of particles consisting of the B particle and all the Nbbackground particles, we write down an equation for the dynamics of only the B particle in a background of the small particles. The effect of the small particles on the dynamics of the B particle now only appears via the drag force –ζand the random force, a description of their detailed dynamics no longer is needed nor, in fact, is possible in this treatment. Further, the drag force –ζis calculated by the theory of macroscopic hydrodynamics and the random force is assumed to be a Gaussian random variable, that is, is treated stochastically and only its statistical properties are needed. Thus, we have reduced the description of the dynamics of an Nb+ 1 body problem to that of a one-body problem. However, we have lost information as we now no longer are able using this approach to follow the dynamics of the Nbbath particles as we have averaged-out or coarse-grained over their motion.

This approach pioneered by Langevin 1 has been made formal, vastly extended, and has been shown to be applicable to any dynamical variable. The basic ideas used to generalise the traditional Langevin equation are:

1. First define the dynamical variables of interest, for example, the velocities of particles.

2. Write down a set of coupled equations of motion for these variables in operator form, for example, for the velocities of some particles of the system.

3. Rewrite these equations of motion so as to project out the variables in which we are interested by use of projection operators.

This projecting out averages over the motion of the other particles in the system and is a coarse graining of the equations of motion which provides a description only of the variables which are not averaged over, thus, losing information about the dynamics of the system. Furthermore, this averaging or projection leaves us with terms, “random forces” in the resulting equations of motion for the desired variables which we only have limited information about. These terms are deterministic in the sense that, if we were to go back to the full equations of motion we would know them exactly but this would defeat the purpose of deriving the projected equations of motion. Thus, the “random forces” must, of necessity, be treated in practice as stochastic or random variables about which we only know their statistical properties.

The result of this process is as in the LE approach, one equation for each dynamical variable chosen rather than many coupled equations with which we started. There may still, of course, be many coupled equations but many fewer than we started with and which explicitly involve many less variables than we started with. For example in the Mori–Zwanzig approach 2 we may derive a single, exact equation for the velocity of a typical particle of each species and not one equation for every particle in the system that would constitute the normal kinetic theory approach. However, it should be emphasised that the resulting equations are still exactly equivalent to the original coupled equations on which they are based and we have not eliminated the coupling of the dynamics of each particle. This coupling will be shown to be represented by a Kernel (or memory function) and a “random force” appearing in these equations. Thus, until we are able to calculate or approximate these two terms we have not achieved a solution to the problem of describing the dynamical properties of the system and their time evolution.

There are, however, advantages to this approach some of which are

1. There is one equation per “species” to be solved.

2. These equations are exact and entirely equivalent to the original equations of motion.

3. These equations may be readily used to construct approximate equations of motion.

4. Equations of motion for time-correlation functions, which can be used to calculate linear transport coefficients and scattering functions, may be directly derived from these basic equations.

5. The form of the memory function often gives us physical insight into the processes involved in the relaxation of a variable to equilibrium.

In order to carry out the above derivations we must first give an outline of the classical mechanical description of dynamics mentioned above and then establish the basis for deriving the generalised Langevin description. For completeness we will also provide an outline of how such dynamical information may be used to calculate the mechanical and non-mechanical properties of a many-body system. In subsequent chapters equations of the GLE type will be derived, basic applications given and numerical schemes outlined for solving them which are analogous to the MD method based on Newton’s equations of motion.

1.2 THE MECHANICAL DESCRIPTION OF A SYSTEM OF PARTICLES

The classical mechanical description of the instantaneous mechanical state of system only requires the specification of the set of positions and momenta for all the Nparticles in the system. Then the common approach to describing the time evolution of these variables, the dynamics, of such a many-body system is by use of a coupled set of differential equations describing the detailed time evolution of the position and momentum of each particle. Usually this time evolution is described by Newton’s equations of motion written in terms of Cartesian co-ordinates but other equations of motion may be used such as the Lagrange or Hamilton equations and various co-ordinate and momenta schemes used as appropriate. However, we will not give a detailed description of these aspects of mechanics as they are very well known.3 It does, however, seem wise to comment on the description of a many-body system as given by this classical treatment, show how its output may be used to calculate the macroscopic properties of the system, to discuss the limitations it imposes on the calculation of these properties and what extensions are needed to this basic approach to treat some properties and some types of systems. These points need to be clarified before transforming this familiar Newtonian mechanical description of a system into the less familiar generalised Langevin one.

To reiterate as the classical mechanical state of a system at a particular time is specified by giving the value of the positions of the particles and their momenta (represented by the symbol Γ) then all mechanical properties may similarly be expressed in terms of these...

Produkt-Details

• Erscheint lt. Verlag: 1.12.2007
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Atom- / Kern- / Molekularphysik
• Themenwelt: Technik
• ISBN-10: 0-08-046792-X / 008046792X
• ISBN-13: 978-0-08-046792-4 / 9780080467924