Percolation Theory Using Python.

• Superior document: Lecture Notes in Physics Series ; v.1029
• Author: Malthe-Sørenssen, Anders.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2024.; ©2024.
• Year of Publication: 2024
• Edition: 1st ed.
• Language: English
• Series: Lecture Notes in Physics Series
• Physical Description: 1 online resource (221 pages)

Table of Contents

Intro -- 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&gt -- pc and L ξ -- Conductance of the Spanning Cluster -- Conductivity for p&gt -- 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&lt -- pc -- Diffusion at p = pc -- Diffusion for p&gt -- 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.