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

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

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ISBN: 9780470046074 | 0470046074
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Copyright: 6/30/2006

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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

CHAPTER 1 DIELECTIC RELAXATION PHENOMENA IN COMPLEX MATERIALS

(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

(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

(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

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

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