Applications of Percolation Theory

Preface to the Second Edition
Preface to the First Edition
Contents
1 Macroscopic Connectivity as the Essential Characteristic of Heterogeneous Systems
1.1 Introduction
1.2 What is Percolation?
1.3 The Scope of the Book
1.4 Further Reading
2 Percolation Theory: Classical and Poor Man's
2.1 Introduction
2.2 Historical Origin of Percolation Theory
2.3 Random Bond and Site Percolation
2.4 Percolation Thresholds
2.5 Bicontinuous Materials, Phase-Inversion Symmetry, and Percolation
2.6 Generating a Single Percolation Cluster: The Leath–Alexandrowicz Algorithm
2.7 The Newman–Ziff Algorithm
2.8 Percolation Properties
2.8.1 Connectivity Properties
2.8.2 Flow and Transport Properties
2.8.3 The Structure of the Sample-Spanning Cluster
2.9 Universal Power Laws for Percolation Properties
2.9.1 Connectivity Properties
2.9.2 Flow and Transport Properties
2.10 Scale Dependence of Percolation Properties
2.11 Finite-Size Scaling
2.12 Experimental Measurement of Percolation Properties
2.13 Poor Man's Percolation Theory: Effective-Medium Approximation
3 Extensions of the Classical Percolation Model
3.1 Introduction
3.2 Continuum Percolation
3.2.1 Percolation Threshold of Continua
3.2.2 The Ornstein–Zernike Formulation
3.2.3 Percolation Thresholds: Materials with Very Low or High Thresholds
3.2.4 Differences Between Lattice and Continuum Percolation
3.2.5 Boiling: An Application of Continuum Percolation
3.3 Correlated Percolation
3.3.1 Percolation with Short-Range Correlations
3.3.2 Percolation with Long-Range Correlations
3.3.3 Correlated-Site Percolation and Unusual Properties of Low-Temperature Water
3.3.4 Site-Bond Correlated Percolation and Gelation
3.4 Bootstrap Percolation
3.4.1 k-Core Percolation
3.5 Invasion Percolation
3.6 Explosive Percolation
3.7 Directed Percolation
3.8 Dynamic Percolation
3.9 Percolation Model of Galactic Structures
3.10 Vector Percolation
4 Characterization of Porous Media and Materials
4.1 Introduction
4.2 Diagenetic Processes and Formation of Rock
4.3 Geometrical-Percolation Models of Diagenetic Processes
4.4 Percolation Model of the Tortuosity
4.4.1 Tortuosity and Continuum Percolation
4.4.2 Percolation Models for Tortuosity of Saturated and Unsaturated Porous Media
4.5 Characterization of Pore-Space Geometry
4.5.1 The Young–Laplace Equation
4.5.2 Contact Angle, the Washburn Equation, and Capillary Pressure
4.5.3 Mercury Porosimetry
4.5.4 Invasion and Random Percolation Models
4.6 Sorption in Porous Media
4.6.1 Percolation Models of Sorption
4.6.2 Determining Connectivity of Porous Materials Using Percolation
5 Connectivity of Fracture and Fault Networks
5.1 Introduction
5.2 Why Should Percolation be Relevant to Fractures?
5.3 Self-Similar Fractal Structures
5.4 Experimental Evidence for Self-Similarity of Fracture Networks and Relevance of Percolation
5.5 Excluded Area and Excluded Volume
5.6 Models of Fracture Networks and Their Percolation Threshold
5.7 Two-Dimensional Models
5.8 Three-Dimensional Models
5.9 Percolation Threshold of Fracture Networks
6 Earthquakes, Critical Phenomena, and Percolation
6.1 Introduction
6.2 Earthquakes as a Critical Phenomenon
6.3 Spatial Distribution of Earthquakes Hypocenters and its Relation …
6.4 Microseismicity and Percolation
6.5 The Gutenberg–Richter Law and its Relation with Percolation
6.6 Percolation Model of Regional Seismicity
7 Flow and Transport Properties of Porous Materials
7.1 Introduction
7.2 Percolation, Poor Man's Percolation, or Critical-Path Analysis?
7.3 Selecting the Correct Percolation Model of the Unsaturated Zone in Soil
7.4 Diffusivity
7.4.1 Percolation Approach
7.4.2 Poor Man's Percolation: Effective-Medium Approximation
7.4.3 Effective-Medium Approximation and Percolation
7.5 Permeability
7.5.1 Poor Man's Percolation: Effective-Medium Approximation
7.5.2 Critical-Path Analysis
7.5.3 Percolation-CPA Models of Hydraulic Conductivity and Air Permeability
7.6 Electrical Conductivity
7.6.1 Effective-Medium Approximation
7.6.2 Combining Effective-Medium Approximation and Percolation
7.6.3 Critical-Path Analysis
7.6.4 Renormalization Group Method
7.6.5 Renormalized Effective-Medium Approximation
7.6.6 Percolation and CPA models of Electrical Conductivity and Electrokinetic Coupling
7.7 Thermal Conductivity
7.7.1 Percolation Model
7.7.2 Effective-Medium Approximation
7.7.3 Percolation-Modified Effective-Medium Approximation
7.7.4 Universality Across Flow and Transport Properties
8 Mixing and Dispersion in Flow Through Porous Media
8.1 Introduction
8.1.1 The Phenomenon of Dispersion
8.2 Mechanisms of Dispersion
8.3 The Convective–Diffusion Equation
8.4 Taylor–Aris Dispersion in a Capillary Tube
8.5 Classification of Dispersion Regimes in Porous Media
8.6 Percolation and Pore-Network Models
8.7 Effect of Percolation and Connectivity
8.7.1 Scaling Theory
8.8 Critical-Path Analysis
8.9 Comparison with Experiments
9 Multiphase Fluid Flow in Porous Media
9.1 Introduction
9.2 Random Percolation Model
9.3 Computing the Relative Permeabilities with Random Percolation Model
9.3.1 Use of the Bethe Lattices
9.3.2 Poor Man's Percolation: Effective-Medium Approximation
9.4 Random Site-Correlated Bond Percolation Models
9.5 Invasion Percolation
9.5.1 Comparison with Experimental Data
9.6 Flow of Thin Wetting Films in Pores
9.7 Invasion Percolation with Two Invaders and Two Defenders
9.8 Random Percolation with Trapping
9.9 Roughening and Pinning of a Fluid Interface
9.10 Comparison with Experimental Data and Relation with Directed Percolation
9.11 Finite-Size Effects and Devil's Staircase
9.12 Displacement Under the Influence of Gravity: Gradient Percolation
9.12.1 Comparison with Experiments
9.13 Immiscible Displacements at Finite Capillary Numbers …
9.14 Evaporation and Drying in Porous Media
9.14.1 Pore-Network Simulation
9.14.2 Scaling Theory of Drying
10 Percolation in Evolving Media
10.1 Introduction
10.2 Noncatalytic Gas–Solid Reactions with Fragmentation
10.2.1 The Reaction-Limited Regime
10.2.2 Diffusion-Limited Regime
10.3 Percolation Models of Coal Gasification with Fragmentation
10.3.1 The Reaction-Limited Regime
10.3.2 Hybrid Percolation-Continuum Models of Diffusion-Limited Regime
10.3.3 Pore-Network Models
10.4 Noncatalytic Gas–Solid Reactions with Pore Blocking
10.5 Percolation Models of Catalyst Deactivation
10.5.1 Reaction-Limited Deactivation
10.5.2 Diffusion-Limited Deactivation
10.5.3 Catalyst Deactivation at the Reactor Scale
10.6 Reaction Kinetics and Diffusion-Controlled Annihilation
10.7 Clogging of Porous Media by Precipitation of Particles …
11 Vector Percolation and Rigidity of Materials
11.1 Introduction
11.2 Derivation of Elasstic Networks From Continuum Elasticity
11.3 The Born Model
11.4 The Central-Force Networks
11.4.1 Elastic Networks in Biological Materials
11.5 Static and Dynamic Rigidity and Floppiness of Networks
11.6 Rigidity Percolation
11.7 Elastic and Superelastic Percolation Models
11.7.1 Numerical Simulation and Finite-Size Scaling
11.7.2 The Correlation Length of Rigidity Percolation
11.7.3 Finite-Size Scaling Analysis
11.8 The Force Distribution
11.9 Determination of the Elastic Percolation Threshold
11.9.1 Estimating the Percolation Threshold by Moments of the Force Distribution
11.9.2 Constraint-Counting Method
11.9.3 The Pebble Game
11.10 Elastic Percolation Networks with Bond-Bending Forces
11.10.1 The Percolation Threshold
11.10.2 The Force Distribution
11.11 Nature of Phase Transition in Elastic Percolation Netwoks
11.12 Scaling Properties of the Elastic Moduli
11.13 Poor Man's Percolation: Effective-Medium Approximation
11.13.1 The Born Model
11.13.2 Central-Force Percolation
11.13.3 The Bond-Bending Models
11.13.4 Filamentous Networks
11.14 Critical-Path Analysis
11.15 Fixed Points of Elastic Percolation and University of the Poisson's Ratio
12 Transport Properties of Composite Materials
12.1 Introduction
12.2 Electrical Conductivity of Powders and Polymer Composites
12.2.1 Powders
12.2.2 Polymer Composites
12.2.3 Polymer–Graphene Composites
12.2.4 Polymer–Carbon Nanotube Composites
12.2.5 Effect of Thickness and Particle-Size Distribution
12.3 Metal–Insulators Composites
12.4 Tunneling Versus Percolation
12.5 AC Conductivity and Dielectric Properties of Heterogeneous Composites
12.5.1 AC Conductivity
12.5.2 Dielectric Constant
12.5.3 Experimental Verification
12.6 Hall Conductivity
12.6.1 Poor Man's Percolation: Effective-Medium Approximation
12.6.2 Scaling Properties
12.6.3 Comparison with Experimental Data
12.7 Mechanical Properties of Heterogeneous Materials
12.7.1 Foams
12.7.2 Porous Materials
12.7.3 Superrigid Materials
12.8 Percolation Aspects of Classical Superconductivity
12.8.1 Magnetoconductivity
12.8.2 Magnetic Properties
12.8.3 Comparison with Experimental Data
12.9 Thermal Percolation
13 Rheology and Elastic Properties of Network Glasses, Branched Polymers, and Gels
13.1 Introduction
13.2 Connectivity of Network Glasses and Rigidity Transition
13.2.1 Experimental Confirmation of the Phillips–Thorpe Theory
13.2.2 Network Glasses with High Coordination Numbers
13.2.3 Network Glasses with Dangling Atoms
13.2.4 Stress Transition in Network Glasses
13.3 Branched Polymers and Gels
13.3.1 Percolation Model of Polymerization and Gelation
13.3.2 Morphology of Branched Polymers and Gels
13.4 Elastic Moduli of Critical Gels
13.4.1 The Spectrum of the Relaxation Times
13.4.2 Experimental Confirmation
13.4.3 Chemical Gels
13.4.4 Enthalpic Versus Entropic Elasticity
13.4.5 Physical Gels
13.5 Viscosity of Near Critical Gelling Solutions
14 Vibrational Density of States of Heterogeneous Materials
14.1 Introduction
14.2 Scalar Percolation Approximation for the Density of States
14.2.1 Poor Man's Percolation: Effective-Medium Approximation
14.2.2 Scaling Theory of Phonons and Fractons
14.2.3 Crossover from Phonons to Fractons
14.2.4 Experimental and Numerical Confirmation of the Scalar Approximation
14.3 Vector Percolation and Vibrational Density of States
14.3.1 Scaling Theory
14.3.2 Crossover Between Scalar and Vector Models of Density of States
14.3.3 Experimental and Numerical Confirmation of Vector Percolation Model of Density of States
15 Hopping Conductivity of Heterogeneous Materials
15.1 Introduction
15.2 The Miller-Abrahams Network
15.3 The Symmetric Hopping Model
15.3.1 Exact Solution for One-Dimensional Materials
15.3.2 Exact Solution for Bethe Lattices
15.3.3 Poor Man's Percolation: Effective-Medium Approximation
15.4 Poor Man's Percolation: EMA Derivation of Mott's Formula
15.5 Critical-Path Analysis
15.5.1 Effect of a Variable Density of States
15.5.2 Effect of Coulomb Interactions
15.5.3 Comparison with Experimental Data
15.6 Effect of Fractal Morphology on Hopping Conductivity
15.7 Universality of Frequency-Dependent Hopping Conductivity
16 Applications of Invasion Percolation
16.1 Introduction
16.2 Invasion Percolation
16.3 Extensions of Invasion Percolation
16.3.1 Self-Avoiding and Loopless Compressible Invasion Percolation
16.3.2 Gradient Invasion Percolation
16.3.3 Thermal Invasion Percolation
16.4 Dynamics of Invasion Percolation: The Relation with Self-Organized Criticality
16.5 Applications
16.5.1 Modeling of Stream Networks
16.5.2 Healthy and Cancerous Vascular Networks
16.5.3 Simulating Spin Systems Near and Away from Criticality
16.5.4 Optimization Problems
16.5.5 The Queuing Problem
16.5.6 Social Dynamics
17 Percolation in Random Graphs and Complex Networks
17.1 Introduction
17.2 Erdös-Rényi Graph
17.2.1 Properties of Erdös-Réyni Graph
17.2.2 Percolation in Erdös-Réyni Graph
17.3 Small-World and Scale-Free Networks
17.4 Generation of Scale-Free Networks
17.5 Percolation Properties
18 Percolation in Biological Systems
18.1 Introduction
18.2 Antigen–Antibody Reactions and Aggregation
18.3 Network Formation on Lymphocyte Membranes
18.4 Percolation in Immunological Systems
18.5 Percolation Conductivity in Biological Materials
18.6 Neuromorphic Computing
18.7 Sensory Transmission and Loss of Consciousness
18.8 Actomyosin Networks
18.9 Percolation Transition in the Mutation-Propagating Module Cluster …
18.10 Essential Nodes in Brain: Optimal Percolation
18.11 Flexibility of Thought in High Creative Individuals
18.12 Cardiac Fibrosis and Arrhythmia
18.13 Connectivity of Temporal Hierarchical Mobility Networks During COVID-19
18.14 Protein Structure
18.15 Molecular Motors and Mechanical Properties of Cells and Active Gels
18.16 Percolation and Evolution
18.17 Microbial Communications
19 Percolation Theory, Ecology, Hydrology, and Geochemistry
19.1 Introduction
19.2 Chemical Weathering
19.3 The Water Cycle
19.4 Scaling Theory of Low-Temperature Geochemistry
19.5 Weathering Rinds
19.6 Vegetation Growth
19.7 River Drainage
19.8 Water Balance
19.9 Variability and Climate
19.10 Plants
19.11 Improving Production of Plants with Plague Susceptibility
19.12 Spread of Fungi in Soil
19.13 Pattern of Tropical Deforestation
20 Explosive Percolation and Its Applications
20.1 Introduction
20.2 Explosive Percolation
20.2.1 Explosive Percolation Through the Achlioptas Process
20.2.2 Explosive Percolation Through Achlioptas Process and the Product Rule
20.2.3 Explosive Percolation Through the BFW Model
20.2.4 Explosive Percolation by the n-Edge Model
20.2.5 Explosive Percolation by the Spanning Cluster—Avoiding Rule
20.3 Continuous Versus Discontinuous Phase Transition: Powder Keg …
20.3.1 The Importance of Selecting the Correct Order Parameter
20.3.2 Reversible Versus Irreversible Phase Transition
20.4 Applications of Explosive Percolation
20.4.1 Electric Breakdown
20.4.2 Formation of Bundles of Nanotubes with Uniform Size
20.4.3 Crackling Noise in Fractional Percolation
20.4.4 Diffusion-Limited Cluster Aggregation
20.4.5 Emergence of Molecular Life
20.4.6 Evolutionary Process in Protein Networks
20.4.7 Global Trade Network and Transportation
20.4.8 Social Networks
20.4.9 Explosive Immunization
20.4.10 Optimal Dismantling and Explosive ``Death''
20.4.11 Spreading of Information
21 Directed Percolation and Its Applications
21.1 Introduction
21.2 Statistics of Vortex Filaments and Percolation
21.3 Directed Percolation
21.4 General Model of Directed Percolation and Random Resistor–Diode Networks
21.5 Directed Percolation and Transition Between Laminar and Turbulent Flows
21.5.1 Numerical Simulation
21.5.2 Experimental Confirmation
21.6 Turbulent Flow and Ecological Collapse
21.7 Catalytic Reactions
21.8 Depinning of Interface in Two-Phase Flow in Porous Media
21.9 Flowing Sand
21.10 Neuronal Avalanche in Brain
21.11 Satellite Networks
22 Percolation in Large-Scale Systems
22.1 Introduction
22.2 Mobile Ad Hoc Networks
22.3 Topography of Earth and Moon
22.4 Ancient Sea Level on Mars
22.5 Forest Fires
22.5.1 Self-Organized Critical Model of Forrest Fires
22.6 Percolation in Sea Ice
22.7 Lifecycle of Industrial Products and Consumer Demand
22.8 Spreading of Technological Innovations
22.9 Social Percolation
Appendix References
Index