Fractals, Diffusion and Relaxation in Disordered Complex Systems, Volume 133, 2 Volumes

by Kalmykov, Yuri P.; Coffey, William T.; Rice, Stuart A.

• ISBN13: 9780470046074
• ISBN10: 0470046074
• Edition: 11th
• Format: Hardcover
• Copyright: 2006-06-30
• Publisher: Wiley-Interscience

Summary

This series provides the chemical physics field with a forum for critical, authoritative evaluations of advances in every area of the discipline. Fractals, Diffusion, and Relaxation in Disordered Complex Systems is a special guest-edited, two-part volume of Advances in Chemical Physics that continues to report recent advances with significant, up-to-date chapters by internationally recognized researchers. This special volume includes chapters on: * Dielectic Relaxation Phenomena in Complex Materials * Evolution of the Dynamic Susceptibility in Super-Cooled Liquids and Glasses * Slow Relaxation, Anomalous Diffusion, and Aging in Equilibrated or Non-equilibrated Environments * Aging and Non-Ergodicity Behavior of Blinking Quantum Dot * The Continuous Time Random Walk Versus the Generalized Master Equation

Author Biography

William Coffey has held research positions at the School of Theoretical Physics, Dublin Institute for Advanced Studies, and at the University of Salford. He was appointed Lecturer at the University of Dublin, Trinity College, in 1977, elected a Fellow of the College in 1981, and appointed a Professor of Engineering Sciences in 1985. Renowned for his work in the theory of Brownian motion and molecular diffusion Dr. Coffey has been the recipient of numerous honors, including Fellow of the American Physical Society, Member of the Royal Irish Academy, and Docteur Honoris causa (Université de Perpignan, France). Professor Coffey is the author of many books and more than one hundred and fifty papers. In particular, he is the coauthor of two previous Wiley books, Molecular dynamics (with M. W. Evans, G. J. Evans, and P. Grigolini) and Molecular Diffusion and Spectra (with M. W. Evans and P. Grigolini).

Yuri Kalmykov is Professor of Physics at the University of Perpignan. Dr. Kalmykov has also been a Visiting Scientist at Trinity College, Dublin, and a Visiting Professor at Queen's University of Belfast, UK. The area of his research interests is non-equilibrium statistical mechanics, dielectric and Kerr-effect relaxation in gaseous and liquid dielectrics, magnetic relaxation of ferrofluids, relaxation processes in complex systems, etc. Dr. Kalmykov is a Fellow of the Institute of Physics, UK, and author of several books and over two hundred research articles that have appeared in numerous scientific journals including the Journal of Chemical Physics, the Physical Review (A, B, and E) and Physical Review Letters.

Table of Contents

CHAPTER 1 DIELECTIC RELAXATION PHENOMENA IN COMPLEX MATERIALS	1	(126)
By Yuri Feldman, Alexander Puzenko, and Yaroslav Ryabov
I. Introduction
II. Dielectric Polarization, Basic Principles
A. Dielectric Polarization in Static Electric Fields
1. Types of Polarization
B. Dielectric Polarization in Time-Dependent Electric Fields
1. Dielectric Response in Frequency and Time Domains
C. Relaxation Kinetics
III. Basic Principles of Dielectric Spectroscopy and Data Analyses
A. Basic Principles of the BDS Methods
B. Basic Principles of the TDS Methods
1. Experimental Tools
2. Data Processing
C. Data Analysis and Fitting Problems
1. The Continuous Parameter Estimation Problem
2. dc-Conductivity Problems
3. Continuous Parameter Estimation Routine
4. Computation of the dc-Conductivity Using the Hilbert Transform
5. Computing Software for Data Analysis and Modeling
IV. Dielectric Response in Some Disordered Materials
A. Microemulsions
B. Porous Materials
1. Porous Glasses
2. Porous Silicon
C. Ferroelectric Crystals
D. H-Bonding Liquids
V. Cooperative Dynamic and Scaling Phenomena in Disordered Systems
A. Static Percolation in Porus Materials, Fractal Concept, and Porosity Determination
1. Porous Glasses
2. Porous Silicon
B. Dynamic Percolation in Ionic Microemulsions
1. Dipole Correlation Function for the Percolation Process
2. Dynamic Hyperscaling Relationship
3. Relationship Between the Static and Dynamic Fractal Dimensions
C. Percolation as Part of "Strange Kinetic" Phenomena
D. Universal Scaling Behavior in H-Bonding Networks
1. Glycerol-Rich Mixtures
2. Water-Rich Mixtures
E. Liquid-like Behavior in Doped Ferroelectric Crystals
F. Relaxation Kinetics of Confined Systems
1. Model of Relaxation Kinetics for Confined Systems
2. Dielectric Relaxation of Confined Water
3. Dielectric Relaxation in Doped Ferroelectric Crystal
4. Possible Modifications of the Model
5. Relationships Between the Static Properties and Dynamics
G. Dielectric Spectrum Broadening in Disordered Materials
1. Symmetric Relaxation Peak Broadening in Complex Systems
2. Polymer-Water Mixtures
3. Microcomposite Material
VI. Summary
Acknowledgments
References
CHAPTER 2 EVOLUTION OF THE DYNAMIC SUSCEPTIBILITY IN SUPERCOOLED LIQUIDS AND GLASSES	127	(130)
By Thomas Blochowicz, Alexander Brodin, and Ernst Rössler
I. Introduction
II. Selected Methods to Study Molecular Reorientation Dynamics
A. Molecular Reorientation: Correlation Function, Spectrum, and Susceptibility
B. Dielectric Spectroscopy
C. Depolarized Light Scattering
D. Nuclear Magnetic Resonance Spectroscopy
1. NMR Time Windows and Relevant Spin–Lattice Interactions
2. Spin–Lattice Relaxation and Line-Shape Analysis
3. Stimulated Echo Experiments and Two-Dimensional NMR
III. Some Comments on Theoretical Approaches to the Glass Transition Phenomenon
A. General Remarks
B. Mode Coupling Theory
IV. Experimental Results of Molecular Glass Formers at T > Tg
A. General Overview
1. Evolution of the Dynamic Susceptibility
2. Time Constants and Decoupling Phenomenon
B. The High-Temperature Regime (T >> Tg)
C. The Low-Temperature Regime (T >~ Tg)
1. Evolution of the Excess Wing
2. Glass Formers with a β-Peak
3. Mechanism of Molecular Reorientation—Results from NMR
D. Temperature Dependence of the Nonergodicity Parameter
E. Tests of Mode Coupling Theory
1. The Asymptotic Scaling Laws
2. Beyond the Asymptotic Scaling Laws
F. Summary: The Evolution of the Dynamic Susceptibility above Tg
V. Experimental Results of Molecular Glasses (T less than Tg)
A. Nearly Constant Loss and Low-Temperature Properties of Molecular Glasses
B. The β-Process—Results from NMR
VI. Conclusions
Acknowledgments
References
CHAPTER 3 SLOW RELAXATION, ANOMALOUS DIFFUSION, AND AGING IN EQUILIBRATED OR NONEQUILIBRATED ENVIRONMENTS	257	(70)
By Noëlle Pottier
I. Introduction
II. Statistical Mechanics of Systems in Contact with Environments
A. The Caldeira–Leggett Model
1. The Caldeira–Leggett Hamiltonian
2. The Coupled Particle Equation of Motion
3. The Statistical Properties of the Random Force
B. Phenomenological Modeling of Dissipation
1. The Spectral Density of the Coupling
2. The Fluctuation–Dissipation Theorem of the Second Kind
3. The Generalized Classical or Quantal Langevin Equation
C. Ohmic Dissipation
1. The Ohmic Spectral Density
2. The Infinitely Short Memory Limit
III. Time-Domain Formulation of the Fluctuation–Dissipation Theorem
A. General Time-Domain Formulation
1. The Dissipative Part of the Response Function in Terms of the Symmetrized Correlation Function
2. The Symmetrized Correlation Function in Terms of the Dissipative Part of the Response Function
B. The Classical Limit
1. The Dissipative Part of the Response Function in Terms of the Derivative of the Symmetrized Correlation Function
2. The Classical FDT
C. The Extreme Quantum Case
1. The Zero-Temperature FDT
2. The Zero-Temperature Analytic Signal
3. Other Representations of the Analytic Signal
IV. Aging Effects in Classical or Quantal Brownian Motion
A. Aging Effects in Overdamped Classical Brownian Motion
1. The Displacement Response and Correlation Functions
2. The Fluctuation–Dissipation Ratio
B. Aging Effects in the Langevin Model
1. The Velocity Correlation Function
2. The Displacement Response and Correlation Functions
3. The Fluctuation–Dissipation Ratio in the Langevin Model
4. Analysis in Terms of the Time-Dependent Diffusion Coefficient
5. Behavior of the Fluctuation–Dissipation Ratio and of the Effective Temperature in the Langevin Model
C. Aging Effects in Quantum Brownian Motion
1. The Velocity Correlation Function
2. The Quantum Time-Dependent Diffusion Coefficient
3. The Displacement Response and Correlation Functions
4. Relation Between the Displacement Response Function and the Time-Dependent Diffusion Coefficient
5. The Modified Quantum Fluctuation–Dissipation Theorem
6. Determination of the Effective Temperature
7. The Effective Temperature in the Ohmic Model
8. Discussion
V. Aging Effects in Classical or Quantal Anomalous Diffusion
A. Non-Ohmic Noise and Friction
B. Non-Ohmic One- and Two-Time Dynamics
1. Mittag–Leffier Relaxation of an Initial Velocity Fluctuation
2. The Velocity Correlation Function
3. The Particle Coordinate and Displacement
4. The Time-Dependent Quantum Generalized Diffusion Coefficient
C. Non-Ohmic Classical Aging Effects
D. Non-Ohmic Quantum Aging Effects
VI. Anomalous Diffusion in Out-of-Equilibrium Environments
A. Diffusion in a Thermal Bath
1. The Fluctuation–Dissipation Theorems
2. Regression Theorem
B. Diffusion in an Out-of-Equilibrium Environment
1. Equation of Motion of a Particle Linearly Coupled to an Out-of-Equilibrium Environment
2. Stationary Medium Case
3. Thermal Bath Case
C. Out-of-Equilibrium Linear Response Theory
1. Age- and Frequency-Dependent Response Functions
2. Quasi-Stationary Regime: Introduction of an Effective Temperature
D. Effective Temperature in an Out-of-Equilibrium Medium: The Link with the Kubo Formulas for the Generalized Susceptibilities
1. Definition of the Effective Temperature
2. The Modified Kubo Formula for the Mobility
3. The Modified Kubo Formula for the Generalized Friction Coefficient
E. The Out-of-Equilibrium Generalized Stokes–Einstein Relation and the Determination of the Effective Temperature
1. The Out-of-Equilibrium Generalized Stokes–Einstein Relation
2. The Effective Temperature
F. The Effective Temperature in a Model with Power-Law Behaviors
1. The Generalized Friction Coefficient and the Mobility
2. The Mean-Square Displacement
3. Determination of the Effective Temperature
G. Discussion
VII. Summary
Acknowledgments
References
Appendix: Some Useful Fourier Transforms and Convolution Relations
A.1. Fourier Transforms
A.2. Convolution Relations
CHAPTER 4 POWER-LAW BUNKING QUANTUM DOTS: STOCHASTIC AND PHYSICAL MODELS	327	(30)
By Gennady Margolin, Vladimir Protasenko, Masaru Kuno, and Eli Barkai
I. Introduction
II. Physical Models
A. Diffusion Model
III. Stochastic Model and Definitions
IV. Aging
A. Mean Intensity of On–Off Process
B. Aging Correlation Function of On–Off Process
C. Case 1
D. Case 2
E. Case 3
V. Nonergodicity
A. Distribution of Time-Averaged Intensity
B. Distribution of Time-Averaged Correlation Function
VI. Experimental Evidence
VII. Summary and Conclusions
Acknowledgments
References
CHAPTER 5 THE CONTINUOUS-TIME RANDOM WALK VERSUS THE GENERALIZED MASTER EQUATION	357
By Paolo Grigolini
I. Introduction
II. Master Equation: A Phenomenological Approach
III. The Generalized Master Equation and the Zwanzig Projection Method
A. The Projection Method
B. Anderson Localization
C. Conclusions
IV. The Continuous-Time Random Walk
V. An Intermittent Dynamic Model
A. Experimental Versus Theoretical Laminar Regions
B. Manneville Map and Its Idealized Version
C. The Nutting Law
D. Final Remarks
VI. An Attempt at Exploring the Superdiffusion Condition
A. The Generalized Central Limit Theorem
B. Toward a Dynamic Derivation of Lévy Processes
C. GME Versus CTRW
D. Gaussian Case
E. Approach Based on Trajectories
F. Multiscaling
G. Conclusions
VII. Dynamic Approach to Anomalous Diffusion: Response to Perturbation
VIII. Dynamic Versus Thermodynamic Approach to Noncanonical Equilibrium
A. Information Approach to Noncanonical Equilibrium
B. Dynamic Approach to Canonical Equilibrium
C. Dynamic Approach to Noncanonical Equilibrium
D. Conclusions
IX. Non-Poisson Dichotomous Noise and Higher-Order Correlation Functions
A. On a Wide Consensus on the General Validity of the DF Assumption
B. Four-Time Correlation Function
X. Non-Poisson Processes and Aging: An Intuitive Approach
XI. Aging: A More Formal Approach
XII. Correlation Functions and Quantum-like Formalism
XIII. Generalized Master Equation of a Given Age
XIV. Limits of the Generalized Master Equations
XV. Non-Poisson and Renewal Processes: A Problem for Decoherence Theory
A. Decoherence Theory
B. Nonordinary Environment
C. Further Problems Caused by Non-Poisson Physics: Transition from the Quantum to the Classical Domain
D. Trajectory and Density Entropies
E. Conclusions
XVI. Non-Poisson Renewal Processes: A Property Conflicting with Modulation Theories
A. Modulation
B. Modulation: No Aging
XVII. An Exhaustive Proposal for Complexity
A. Non-Ohmic Bath
B. Recurrences to x = 0
C. Conclusion
XVIII. Concluding Remarks
A. In Search of a Theory of Complexity
B. Consequences on the Search for Invisible Crucial Events
C. CTRW Versus GME
D. Consequences on the Physics of Blinking Quantum Dots
E. Consequences on Quantum Measurement Processes
F. Conclusion Summary
References
AUTHOR INDEX
SUBJECT INDEX
CHAPTER 6 FRACTAL PHYSIOLOGY, COMPLEXITY, AND THE FRACTIONAL CALCULUS	1	(92)
By Bruce J. West
I. Introduction
II. Scaling in Physiological Time Series
A. Allometric Aggregation Data Analysis
B. Fractal Heartbeats
C. Fractal Breathing
D. Fractal Gait
E. Fractal Neurons
III. Dynamical Models of Scaling
A. Scaling in Time Series
1. Simple Random Walks and Scaling
2. Fractional Random Walks and Scaling
3. Various Inverse
B. Dichotomous Fluctuations with Memory
1. The Exact Solution
2. Early Time Behavior
3. Late Time Behavior
C. Fractals, Multifractals, and Data Processing
1. Multifractal Special
2. Diffusion Entropy Analysis (DEA)
IV. Fractional Dynamics
A. Fractional Calculus
1. Derivative of a Fractal Function
2. Fractional Brownian Motion
B. Fractional Langevin Equations
1. Physical/Physiological Models
C. Fractional Diffusion Equations
D. Langevin Equation with Levy Statistics
V. Summary, Conclusions, and Speculations
References
CHAPTER 7 PHYSICAL PROPERTIES OF FRACTAL STRUCTURES	93	(192)
By Vitaly V. Novikov
I. Introduction
II. Elements of Fractal Theory
A. Continuous, Nowhere Differentiable Functions and Deterministic Fractals
B. Fractal Sets
C. Fractional Hausdorff–Besicovich Dimensions
D. Multifractals
E. Fractal Set Constructed on a Square Lattice
F. Cayley Tree. Ultrametric space
III. Chaotic Structures
A. Percolation Systems
1. Percolation Cluster
2. Critical Indices
3. Renormalization-Group Transformations
4. Physical Properties
B. Fractal Structure Model
1. Properties of Finite Lattices
2. Appendix. The probability functions
IV. Physical Properties
A. Conductivity
1. Maxwell Model
2. The Effective Medium Theory
3. Variational Approach
4. Iterative Averaging Method for Conductivity
B. Frequency Dependence of Dielectric Properties
1. Iterative Averaging Method for Dielectric Properties
C. Galvanomagnetic Properties
1. Iterative Averaging Method for Hall's Coefficient
2. Results and Discussion
3. Appendix. Galvanomagnetic Properties of the Cube Inside a Cube Cell
D. Elastic Properties
1. Iterative Averaging Method for Elastic Properties
2. Results of Calculation
E. Negative Poisson's Ratio
1. Results of Calculations of Poisson's Ratio
F. Frequency Dependence of Viscoelastic Properties
1. Iterative Averaging Method for Viscoelastic Properties
2. Results of Calculations for Viscoelastic Media
3. Negative Shear Modulus
4. Appendix. Fractal Model of Shear Stress Relaxation
G. Relaxation and Diffusion Processes
1. Non-Debye Relaxation
2. Anomalous Diffusion
3. Distribution Function of a Brownian Particle with Memory
4. Inertial Effects of a Brownian Particle
5. Appendix. Derivative of Fractal Functions
References
CHAPTER 8 FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS	285	(154)
By William T. Coffey, Yuri P. Kalmykov, and Sergey V. Titov
I. Introduction
II. Microscopic Models for Dielectric Relaxation in Disordered Systems
A. Continuous-Time Random Walk Model
B. Fractional Diffusion Equation for the Cole–Cole Behavior
C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model
D. Fractional Diffusion Equation for the Cole–Davidson and Havriliak–Negami Behavior
E. Fundamental Solution of the Fractional Smoluchowski Equation
III. Fractional Noninertial Rotational Diffusion in a Potential
A. Anomalous Diffusion and Dielectric Relaxation in a Double-well Periodic Potential
B. Fractional Rotational Diffusion in a Uniform DC External Field
C. Fractional Rotational Diffusion in a Bistable Potential with Nonequivalent Wells
1. Matrix Continued Fraction Solution
2. Bimodal Approximation
IV. Inertial Effects in Anomalous Dielectric Relaxation
A. Metzler and Klafter's Form of the Fractional Klein–Kramers Equation
B. Barkai and Silbey's Form of the Fractional Klein–Kramers Equation
C. Inertial Effects in Anomalous Dielectric Relaxation of Linear and Symmetrical Top Molecules
1. Rotators in Space
2. Symmetric Top Molecules
D. Inertial Effects in Anomalous Dielectric Relaxation in a Periodic Potential
E. Fractional Langevin Equation
V. Conclusions
Appendix I: Calculation of Inverse Fourier Transforms
Appendix II: Exact Continued Fraction Solution for Longitudinal and Transverse Responses Appendix III: Dynamic Kerr-Effect Response: Linear Molecules
Appendix IV: Ordinary Continued Fraction Solution for Spherical Top Molecules Appendix V: Kerr-Effect Response
Acknowledgments
References
CHAPTER 9 FUNDAMENTALS OF LÉVY FLIGHT PROCESSES	439	(58)
By Aleksei V. Chechkin, Vsevolod Y. Gonchar, Joseph Klafter, and Ralf Metzler
I. Introduction
II. Definition and Basic Properties of Lévy Flights
A. The Langevin Equation with Lévy Noise
B. Fractional Fokker—Planck Equation
1. Rescaling of the Dynamical Equations
C. Starting Equations in Fourier Space
III. Confinement and Multimodality
A. The Stationary Quartic Cauchy Oscillator
B. Power-Law Asymptotics of Stationary Solutions for c > or equal to 2, and Finite Variance for c > 2
C. Proof of Nonunimodality of Stationary Solution for c > 2
D. Formal Solution of Equation (38)
E. Existence of a Bifurcation Time
1. Trimodal Transient State at c > 4
2. Phase Diagrams for n-Modal States
F. Consequences
IV. First Passage and Arrival Time Problems for Lévy Flights
A. First Arrival Time
B. Sparre Anderson Universality
C. Inconsistency of Method of Images
V. Barrier Crossing of a Lévy Flight
A. Starting Equations
B. Brownian Motion
C. Numerical Solution
D. Analytical Approximation for the Cauchy Case
E. Discussion
VI. Dissipative Nonlinearity
A. Nonlinear Friction Term
B. Dynamical Equation with Lévy Noise and Dissipative Nonlinearity
C. Asymptotic Behavior
D. Numerical Solution of Quadratic and Quartic Nonlinearity
E. Central Part of P(V,t)
F. Discussion
VII. Summary
Acknowledgements
References
VIII. Appendix. Numerical Solution Methods
A. Numerical Solution of the Fractional Fokker—Planck Equation [Eq. (38)] via the Grünwald—Letnikov Method
B. Numerical Solution of the Langevin Equation [Eq. (25)]
CHAPTER 10 DISPERSION OF THE STRUCTURAL RELAXATION AND THE VITRIFICATION OF LIQUIDS	497	(98)
By Kia L. Ngai, Riccardo Casalini, Simone Capaccioli, Marian Paluch, and C.M. Roland
I. Introduction
II. Invariance of the α-Dispersion to Different Combinations fo T and P at Constant τα
A. Molecular Glass-Formers
B. Amorphous Polymers
C. Implication of T-P Superpositioning of the α-Dispersion at Constant &tau:α
D. Invariance of the α-Dispersion to Different T-P at Constant τα Investigated by Techniques Other than Dielectric Spectroscopy
III. Structural Relaxation Properties Are Goverened by or Correlated with the α-Dispersion
IV. The Primitive Relaxation and the Johari-Goldstein Secondary Relaxation
V. The JG Relaxation and Its Connection to Structural Relaxation
A. Pressure Dependence of τJG
B. Invariance of τJG to Variations of T and P at Constant τα
C. Non-Arrhenius Temperature Dependence of τβ Above Tg
D. Increase of τβ on Physical Aging
E. JG Relaxation Strength and Its Mimicry of Enthalph, Entropy, and Volume
F. The Origin of the Dependences of Molecular Mobility on Temperature Pressure, Bolue and Entropy is in τJG to τ0
VI. The Coupling Model
A. Background
B. The Correspondence Between τ0 and &tau:JG
C. Relations Between the Activation Enthalpies of τJG and τα in the Glassy State
D. Explaining the Properties of τJG
E. Consistency with the Invariance of the α-Dispersion of Constanct τα to Different Combination of T and P
F. Relaxation on a Nanometer Scale
G. Component Dynamics in Binary Mixtures
H. Interrelation Between Primary and Secondary Relaxations Ploymerizing Systems
I. A Shortcut to the Consequences of Many-Molecule Dynamics and a Pragmatic Resolution of the Glass Transition Problem
VII. Conclusion
Acknowledgments
References
CHAPTER 11 MOLECULAR DYNAMICS IN THIN POLYMER FILMS	595	(38)
By Friedrich Kremer and Anatoli Serghei
I. Introduction
II. Preparation of Thin Polymer Films
III. Confinement-Induced Mode in Thin Films of cis-1,4-Polyisoprene
A. Experiment
B. Molecular Assignment
C. Simulations
D. Summary: Experiment and Simulations
IV. Confinement Effects on the Molecular Dynamics in Thin Films of Hyperbranched Polyester
V. Confinement Effects in Thin Polystyrene Films
VI. Conclusions Acknowledgments References
AUTHOR INDEX	633	(38)
SUBJECT INDEX	671