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E-raamat: Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems

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(RMIT University, Applied Physics, School of Applied Sciences,
SET Portfolio, Victoria, Australia)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Dec-2006
  • Kirjastus: Elsevier Science Ltd
  • Keel: eng
  • ISBN-13: 9780080467924
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Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal SystemsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-Dec-2006
  • Kirjastus: Elsevier Science Ltd
  • Keel: eng
  • ISBN-13: 9780080467924
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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

Muu info

Unifies many techniques which fall under the heading of the use of Stochastic (Random) Processes
Preface xiii
Notation xv
Background, Mechanics and Statistical Mechanics
1(20)
Background
1(2)
The Mechanical Description of a System of Particles
3(13)
Phase space and equations of motion
7(1)
In equilibrium
7(2)
In a non-isolated system
9(1)
Newton's equations in operator form
10(1)
The Liouville equation
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)
Projection operators
15(1)
Summary
16(2)
Conclusions
18(3)
References
18(3)
The Equation of Motion for a Typical Particle at Equilibrium: The Mori-Zwanzig Approach
21(20)
The Projection Operator
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)
Summary
38(1)
Conclusions
39(2)
References
39(2)
Approximate Methods to Calculate Correlation Functions and Mori-Zwanzig Memory Functions
41(20)
Taylor Series Expansion
41(2)
Spectra
43(1)
Mori's Continued Fraction Method
44(2)
Use of Information Theory
46(2)
Perturbation Theories
48(3)
Mode Coupling Theory
51(1)
Macroscopic Hydrodynamic Theory
52(4)
Memory Functions Calculated by the Molecular-Dynamics Method
56(1)
Conclusions
57(4)
References
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)
Conclusions
69(2)
References
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)
The friction coefficient
98(1)
Self-diffusion coefficient (Ds)
99(3)
The intermediate scattering function F(q,t)
102(1)
Systems in a shear field
102(2)
Summary
104(1)
Conclusions
104(3)
References
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)
PBC in equilibrium
128(1)
PBC in a shear field
129(1)
PBC in elongational flow
129(2)
Conclusions
131(2)
References
131(2)
Brownian Dynamics
133(24)
Fundamentals
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)
Approximate BD schemes
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)
Conclusions
152(5)
References
153(4)
Polymer Dynamics
157(12)
Toxvaerd Approach
159(1)
Direct Use of Brownian Dynamics
160(3)
Rigid Systems
163(3)
Conclusions
166(3)
References
166(3)
Theories Based on Distribution Functions, Master Equations and Stochastic Equations
169(28)
Fokker-Planck Equation
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)
Master Equations
180(9)
The identification of elementary processes
184(2)
Kinetic MC and master equations
186(1)
KMC procedure with continuum solids
187(2)
Conclusions
189(8)
References
191(6)
An Overview
197(4)
Appendix A: Expressions for Equilibrium Properties, Transport Coefficients and Scattering Functions
201(8)
Equilibrium Properties
201(1)
Expressions for Linear Transport Coefficients
202(2)
Scattering Functions
204(5)
Static structure
204(1)
Dynamic scattering
204(2)
References
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)
Basics
223(1)
Short Time Expansions
224(1)
Relative Initial Behaviour of c(t)
224(1)
Appendix G: Time-Reversal Symmetry of Non-Equilibrium Correlation Functions
225(4)
References
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)
Equilibrium Properties
235(1)
Static Structure
235(1)
Time Correlation Functions
236(5)
Self-diffusion
236(1)
Time-dependent scattering
236(1)
Bulk stress
237(1)
Zero time (high frequency) results in the diffusive limit
237(2)
References
239(2)
Appendix K: Monte Carlo Methods
241(8)
Metropolis Monte Carlo Technique
241(2)
An MC Routine
243(6)
References
248(1)
Appendix L: The Generation of Random Numbers
249(2)
Generation of Random Deviates for BD Simulations
249(2)
References
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)
References
253(2)
Appendix N: Calculation of Hydrodynamic Interaction Tensors
255(6)
References
259(2)
Appendix O: Some Fortran Programs
261(40)
Index 301


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