Fractals, Diffusion, and Relaxation in Disordered Complex Systems, Volume 133, Part A

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

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ISBN: 9780471725077 | 0471725072
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Copyright: 6/30/2006

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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.

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 Porous 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 > almost equal to 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—Leffler 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 BLINKING 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

(118)

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

475

(38)

SUBJECT INDEX

513

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Fractals, Diffusion, and Relaxation in Disordered Complex Systems, Volume 133, Part A

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Author Biography

Table of Contents

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