Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices: Physical Interpretat

CHAPTER ONE ................................................................................................................. 1 Introduction ..................................................................................................................... 1 1.1. Formulation of the Transport Problem ................................................................ 4 1.2. Nature of Disorder ............................................................................................... 7 1.3. Relevance to Physical Systems ........................................................................... 10 CHAPTER TWO .............................................................................................................. 15 Theoretical Background ................................................................................................ 15 2.1. Localization Criteria .......................................................................................... 15 2.2. Anderson Model ................................................................................................. 20 2.3. Edwards and Thouless Model ............................................................................ 25 2.4. Scaling Theory ................................................................................................... 29 CHAPTER THREE .......................................................................................................... 32 Spectral Approach ......................................................................................................... 32 3.1. Essence of the Spectral Method ......................................................................... 32 3.2. Simplified Numerical Model ("Toy Model") ..................................................... 40 3.3. Physical Interpretation ...................................................................................... 45 3.4. Scope and Limitations of the Spectral Analysis ................................................. 49 CHAPTER FOUR ............................................................................................................. 51 Delocalization in 2D Lattices of Various Geometries .................................................. 51 4.1. Transport in the Honeycomb, Triangular, and Square Lattices ........................ 51 4.2. Orthogonality Check .......................................................................................... 56 4.3. Equation Fitting ................................................................................................. 57 4.4. Cluster Analysis ................................................................................................. 60 4.5. Comparison Between the Honeycomb and the Triangular Lattices .................. 63 CHAPTER FIVE .............................................................................................................. 65 Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder 65 5.1. Discrete Percolation .......................................................................................... 66 5.2. Formulation of the Transport Problem .............................................................. 69 5.3. Distribution of Variables ................................................................................... 78 5.4. 2D Honeycomb Lattice with Substitutional Disorder ........................................ 83 CHAPTER SIX ................................................................................................................. 89 Transport in 2D Complex Plasma Crystals ................................................................... 89 6.1. Complex Plasma Preliminaries ......................................................................... 90 6.2. Two-Dimensional Dust Crystal Analogue ......................................................... 94 6.3. Transport in the Classical Regime ..................................................