Percolation Theory Using Python

This course-based open access textbook delves into percolation theory, examining the physical properties of random media—materials characterized by varying sizes of holes and pores. The focus is on both the mathematical foundations and the computational and statistical methods used in this field. Designed as a practical introduction, the book places particular emphasis on providing a comprehensive set of computational tools necessary for studying percolation theory. Readers will learn how to generate, analyze, and comprehend data and models, with detailed theoretical discussions complemented by accessible computer codes. The book's structure ensures a complete exploration of worked examples, encompassing theory, modeling, implementation, analysis, and the resulting connections between theory and analysis. Beginning with a simplified model system—a model porous medium—whose mathematical theory is well-established, the book subsequently applies the same framework to realistic random systems. Key topics covered include one- and infinite-dimensional percolation, clusters, scaling theory, diffusion in disordered media, and dynamic processes. Aimed at graduate students and researchers, this textbook serves as a foundational resource for understanding essential concepts in modern statistical physics, such as disorder, scaling, and fractal geometry. Preface Contents 1 Introduction to Percolation 1.1 Basic Concepts in Percolation 1.2 Percolation Probability 1.3 Spanning Cluster 1.4 Percolation in Small Systems 1.5 Further Reading Exercises 2 One-Dimensional Percolation 2.1 Percolation Probability 2.2 Cluster Number Density Definition of Cluster Number Density Measuring the Cluster Number Density Shape of the Cluster Number Density Numerical Measurement of the Cluster Number Density Average Cluster size 2.3 Spanning Cluster 2.4 Correlation Length Exercises 3 Infinite-Dimensional Percolation 3.1 Percolation Threshold 3.2 Spanning Cluster 3.3 Average Cluster Size 3.4 Cluster Number Density Exercises 4 Finite-Dimensional Percolation 4.1 Cluster Number Density Numerical Estimation of n(s,p) Measuring Probability Densities of Rare Events Measurements of n(s,p) When p →pc Scaling Theory for n(s,p) Scaling Ansatz for 1d Percolation Scaling Ansatz for Bethe Lattice 4.2 Consequences of the Scaling Ansatz Average Cluster Size Density of Spanning Cluster 4.3 Percolation Thresholds Exercises 5 Geometry of Clusters 5.1 Geometry of Finite Clusters Analytical Results in One Dimension Numerical Results in Two Dimensions Scaling Behavior in Two Dimensions 5.2 Characteristic Cluster Size Average Radius of Gyration Correlation Length 5.3 Geometry of the Spanning Cluster 5.4 Spanning Cluster Above pc Exercises 6 Finite Size Scaling 6.1 General Aspects of Finite Size Scaling 6.2 Finite Size Scaling of P(p,L) 6.3 Average Cluster Size Measuring Moments of the Cluster Number Density Scaling Theory for S(p,L) 6.4 Percolation Threshold Measuring the Percolation Probability Π(p,L) Measuring the Percolation Threshold pc Finite-Size Scaling Theory for Π(p,L) Estimating pc Using the Scaling Ansatz Estimating pc and ν Using the Scaling Ansatz Exercises 7 Renormalization 7.1 The Renormalization Mapping Iterating the Renormalization Mapping 7.2 Examples Example: One-Dimensional Percolation Example: One-Dimensional Percolation Example: Renormalization on 2d Site Lattice Example: Renormalization on 2d Site Lattice Example: Renormalization on 2d Triangular Lattice Example: Renormalization on 2d Triangular Lattice Example: Renormalization on 2d Bond Lattice Example: Renormalization on 2d Bond Lattice Exercises 8 Subset Geometry 8.1 Singly Connected Bonds 8.2 Self-Avoiding Paths on the Cluster Minimal Path Maximum and Average Path Backbone Scaling of the Dangling Ends Argument for the Scaling of Subsets Blob Model for the Spanning Cluster Mass-Scaling Exponents for Subsets of the Spanning Clusters 8.3 Renormalization Calculation 8.4 Deterministic Fractal Models 8.5 Lacunarity Exercises 9 Flow in Disordered Media 9.1 Introduction to Disorder 9.2 Conductivity and Permeability Electrical Conductivity and Resistor Networks Flow Conductivity of a Porous System 9.3 Conductance of a Percolation Lattice Finding the Conductance of the System Computational Methods Measuring the Conductance Conductance and the Density of the Spanning Cluster 9.4 Scaling Arguments for Conductance and Conductivity Scaling Argument for p>pc and L ξ Conductance of the Spanning Cluster Conductivity for p>pc 9.5 Renormalization Calculation 9.6 Finite Size Scaling Finite-Size Scaling Observations 9.7 Internal Distribution of Currents 9.8 Real Conductivity Exercises 10 Elastic Properties of Disordered Media 10.1 Rigidity Percolation Developing a Theory for E(p,L) Compliance of the Spanning Cluster at p = pc Finding Young's Modulus E(p,L) 11 Diffusion in Disordered Media 11.1 Diffusion and Random Walks in Homogeneous Media Theory for the Time Development of a Random Walk Continuum Description of a Random Walker 11.2 Random Walks on Clusters Developing a Program to Study Random Walks on Clusters Diffusion on a Finite Cluster for p pc Scaling Theory Diffusion on the Spanning Cluster The Diffusion Constant D Exercises 12 Dynamic Processes in Disordered Media 12.1 Introduction 12.2 Diffusion Fronts 12.3 Invasion Percolation Gravity Stabilization Gravity Destabilization References Index