This book provides a graduate-level introduction to three powerful and closely related techniques in condensed matter physics: memory functions, projection operators, and the defect technique. Memory functions appear in the formalism of the generalized master equations that express the time evolution of probabilities via equations non-local in time, projection operators allow the extraction of parts of quantities, such as the diagonal parts of density matrices in statistical mechanics, and the defect technique allows solution of transport equations in which the translational invariance is broken in small regions, such as when crystals are doped with impurities. These three methods combined form an immensely useful toolkit for investigations in such disparate areas of physics as excitation in molecular crystals, sensitized luminescence, charge transport, non-equilibrium statistical physics, vibrational relaxation, granular materials, NMR, and even theoretical ecology. This book explains the three techniques and their interrelated nature, along with plenty of illustrative examples. Graduate students beginning to embark on a research project in condensed matter physics will find this book to be a most fruitful source of theoretical training.

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V. M. (Nitant) Kenkre is Distinguished Professor (Emeritus) of Physics at the University of New Mexico (UNM), USA, retired since 2016. His undergraduate studies were at IIT, Bombay (India) and his graduate work took place at SUNY Stony Brook (USA). He was elected Fellow of the American Physical Society in 1998, Fellow of the American Association for Advancement of Science in 2005 and has won an award from his University for his international work. He was the Director of two Centers at UNM: the Center for Advanced Studies for 4 years and then the Founding Director of the Consortium of the Americas for Interdisciplinary Science for 16 years. He was given the highest faculty research award of his University in 2004 and supervised the Ph.D. research of 25 doctoral scientists and numerous postdoctoral researchers.

Through 270 published papers, his research achievements include formalistic contributions to non- equilibrium statistical mechanics, particularly quantum transport theory, observations in sensitized luminescence and exciton/electron dynamics in molecular solids, and solutions to cross-disciplinary puzzles arising in spread of epidemics, energy transfer in photosynthetic systems, statistical mechanics of granular materials, and the theory of microwave sintering of ceramics.

He has interests in comparative religion, literature and visual art, and has often lectured on the first of these. His most recent coauthored book is Theory of the Spread of Epidemics and Movement Ecology of Animals (Cambridge University Press, 2020). He has also coauthored a book on exciton dynamics (Springer, 1982), coedited another on modern challenges in statistical mechanics (AIP, 2003), and published a book on his poetry entitled Tinnitus, and two on philosophy: The Pragmatic Geeta, and What is Hinduism.

Inhalt

Dedication page

Acknowledgments

Foreword

Authors' Preface

1 The Memory Function Formalism: What and Why 1.1 Introduction to Memory Functions

1.2 An Example of How Memory Functions Arise: the Railway-Track Model

1.3 An Overview of Areas in which the Memory Formalism Helps

2 Zwanzig's Projection Operators: How They Yield Memories

2.1 The Derivation of the Master Equation: a Central Problem in Quantum Statistical Mechanics

2.2 Memories from Projection Operators that Diagonalize the Density Matrix

2.3 Two Simple Examples of Projections and an Exercise 2.3.1 Evolution of a Simple Complex Quantity

2.3.2 Projection Operators for Quantum Control of Dynamic Localization

2.3.3 Exercise for the Reader: the Open Trimer

2.4 What is Missing from the Projection Derivation of the Master Equation

3 Building Coarse-Graining into the Projection Technique

3.1 The Need to Coarse-Grain

3.2 Constructing the Coarse-Graining Projection Operator

3.3 Generalization of the Forster-Dexter Theory of Excitation Transfer

3.4 Obtaining Realistic Memory Functions

3.5 Implementing a General Plan

3.5.1 Example in an Unrelated Area: Ferromagnetism

4 Features of Memory Functions and Relations to Other Entities 4.1 Resolution of the Perrin-Forster-Davydov Puzzle

4.2 Relations Among Theories of Excitation Transfer

4.3 Long-range Transfer Rates as a Consequence of Strong Intersite Coupling

4.4 Connection of Memories to Neutron Scattering and Velocity Auto-Correlation Functions, and Pausing Time Distributions

5 Applications to Experiments: Transient Gratings, Ronchi Rulings, and Depolarization

5.1 Non-drastic Experiments: Fluorescence Depolarization as an Example

5.2 Ronchi Rulings for Measuring Coherence of Triplet Excitons

5.3 Fayer's Transient Gratings: an Ideal Experiment for Measuring Coherence of Singlet Excitons

6 Projection Operators for Various Contexts

6.1 Projections for the Theory of Electrical Resistivity

6.2 Projections that Integrate in Classical Systems

6.2.1 The BBGKY Hierarchy

6.2.2 Torrey-Bloch Equation for NMR Microscopy

6.3 Projections for Quantum Control of Dynamic Localization

6.4 Projections for the Railway-Track Model of Chapter 2

7 Memories and Projections in Nonlinear Equations of Motion

7.1 Extended Nonlinear Systems and the Physical Pendulum

7.2 Nonlinear Waves in Reaction Di_usion systems

7.3 Spatial Memories: Inuence Functions in the Fisher Equation

8 NMR Microsocopy and Granular Compaction

8.1 Pulsed Gradient NMR Signals in Con_ned Geometries

8.2 Analytic Solutions of a Generalized Torrey-Bloch Equation

8.3 Non-local Analysis of Stress Distribution in Compacted Sand

8.4 Spatial Memories and Correlations in the Theory of Granular Materials

9 Projections/Memories for Microscopic Treatment of Vibrational Relaxation

9.1 The Importance of Vibrational Relaxation

9.2 The Montroll-Shuler Equation and its Generalization to the Coherent Domain

9.3 Reservoir E_ects in Vibrational Relaxation

9.4 Approach to Equilibrium of a Simpler System: a Non-Degenerate Dimer

10 The Montroll Defect Technique

10.1 Introduction: Experiments that Modify Substantially

10.2 Overview of the Defect Technique and Simple Cases

10.2.1 Trapping at a Single Site

10.2.2 How Laplace Inversion may be avoided in Some Situations

10.2.3 Trapping at More than 1 Site: Exercise for the Reader

10.3 Coherence E_ects on Sensitized Luminescence

10.4 End-Detectors in a Simpson Geometry 10.5 High Defect Concentration: the _-function Approach

10.6 Periodically Arranged Defects

10.7 Remarks

11 The Defect Technique in the Continuum

11.1 General Discussion

11.2 Higher Dimensional Systems

11.3 A Theory of Coalescence of Signaling Receptor Clusters in Immune Cells

11.4 The Defect Technique with the Smoluchowski Equation 11.5 Momentum-Space Theory of Capture

12 A Mathematical Approach to Non-Physical Defects

12.1 Introduction

12.2 Exciton Annihilation in Translationally Invariant Crystals

12.3 Scattering Function from the Stochastic Liouville Equation with its Terms viewed as Defects

12.4 Transmission of Infection in the Spread of Epidemics

13 Memory Functions from Static Disorder: E_ective Medium Approach

13.1 Introduction

13.2 Various Descriptions of Disorder

13.3 E_ective Medium Approach: Philosophy and Prescription

13.4 Examination of its Validity and Extension of its Applications

14 Concluding Remarks

14.1 What We Have Learnt

Bibliography

Bibliography

Author index

Subject index