We now come to another gem in the Science For Everyone series, Physics and Geometry of Disorder – Percolation Theory by A. L. Efros.
This book is about percolation theory and its various applications, which occur mostly in physics and chemistry. The book is self-sufficient in that it contains chapters on elementary probability theory and Monte Carlo simulation. Most attention is paid to the relationship between the geometrical and physical properties of systems in the vicinity of their percolation thresholds. The theory is applied to examples of impurity semiconductors and doped ferromagnetics, which demonstrate its universality. Although written for students at high schools, the book is very good reading for college students and will satisfy the curiosity of a physicist for whom this will be a first encounter with percolation theory.
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Part I. Site Percolation Problem 14
Chapter 1. Percolation Threshold 14
Two Pundits Shred a Wire Mesh (14).
What Is a Random Variable? (17).
Mean Value and Variance (18).
Why a Large Wire Mesh? (23).
Chapter 2. Basic Rules for Calculating Probabilities Continuous Random Variables 28
Events and Their probabilities (28).
Addition of Probabilities (30).
Multiplication of Probabilities (33).
Percolation Threshold in a 2 x 2 Network (37).
Continuous Random Variables (40).
Percolation Threshold as a Continuous Random Variable (44).
Chapter 3. Infinite Cluster 48
Permanent Magnet (48).
Doped Ferromagnetics (53).
Formation of an Infinite Cluster (56).
Site Percolation Problem Revisited (59).
Clusters at a Low Concentration of Magnetic Atoms(63).
Chapter 4 Solution of the Site Percolation Problem by Monte Carlo Computer Techniques 68
Why Monte Carlo? (68).
What Is the Monte Carlo Method? (70).
How to Think Up a Random Number (74).
The Mid—Square Method (76).
Linear Congruent Method (78).
Determination of Percolation Threshold. by Monte Carlo Simulation on a Computer. Distribution of Blocked and Non-blocked Sites (81).
Search for Percolation Path (85).
Determination of the Threshold (86).
Part II. Various Problems of Percolation Theory and Their Applications
Chapter 5. Problems on Two-Dimensional Lattices 90
We Are Planting an Orchard (the Bond Problem) (90).
Inequality relating x_b to x_s (95).
Covering and Containing; Lattices (98).
“White” Percolation and “Black” Percolation (105).
Dual Lattices (110).
Results for Plane Lattices (116).
Directed Percolation (117).
Chapter 6. Three—Dimensional Lattices and Approximate Evaluation of Percolation Thresholds 120
Three-Dimensional Lattices (121).
Percolation Thresholds for 3D Lattices (126).
Factors Determining Percolation Threshold in the Bond Problem (127).
How to Evaluate Percolation Threshold in the Site Problem (129).
Chapter 7. Ferromagnetics with Long-Range Interaction. The Sphere Problem 135
Ferromagnetics with Long-Range Interaction (136).
The Sphere (Circle) Problem (140).
The Circle (Sphere) Problem Is the Limiting Case Of the Site Problem (144).
Chapter 8. Electric Conduction of Impurity Semiconductors. The Sphere Problem 147
Intrinsic Semiconductors (147).
Impurity Semiconductors (150).
Transition to Metallic Electric Conduction at Increased Impurity Concentrations (158).
The Mott Transition and Sphere Problem (161).
Chapter 9. Various Generalizations of the Sphere Problem 166
Inclusive Figures of Arbitrary Shape (166).
The Ellipsoid Problem (169).
Other Surfaces (173).
Another Experiment at the House Kitchen. The Hard-Sphere Problem (174).
Chapter 10. Percolation Level 179
“The Flood” (179).
How to Construct a Random Function (182).
Analogy to the Site Problem (185).
Percolation Levels in Plane and Three Dimensional Problems (186).
Impurity Compensation in Semiconductors (189).
Motion of a Particle with Nonzero Potential Energy (190).
Motion of an Electron in the Field of Impurities (192).
Part III. Critical Behavior of Various Quantities Near Percolation Threshold. Infinite Cluster Geometry 195
Chapter 11 The Bethe Lattice
Solution of the Site Problem on the Bethe Lattice (200).
Chapter 12. Structure of Infinite Clusters 206
The Shklovskii—de Gennes Model (206).
Role of the System’s Size (210).
Electric Conduction Near Percolation Threshold (215).
Function P (X) Near Percolation Threshold. Role Played by Dead—Ends (219).
Universality of Critical Exponents (222).
Chapter 13. Hopping Electric Conduction 226
Mechanism of Hopping Conduction (227).
Resistor Network (229).
Properties of Resistor NetworK (231).
The Sphere` Problem Revisited (232).
Calculation of Resistivity (233). Discussion of the Result (235).
Chapter 14. Final Remarks 237
Some Applications (237).
What Is Percolation Theory, After All? (240)
Answers and Solutions 242
Chapter 1 (242). Chapter 2 (244). Chapter 3 (246).
Chapter 4 (249). Chapter 5 (250). Chapter 6 (256).
Chapter 7 (257). Chapter 8 (257). Chapter 11 (257).
Chapter 12 (258).