A Size-Perimeter Discrete Growth Model for Percolation Clusters

Bendegúz Dezső Bak,Tamás Kalmár-Nagy,Giacomo Fiumara

doi:10.1155/2021/9919333

Bendegúz Dezső Bak, Tamás Kalmár-Nagy + Show 1 more

Open Access

https://doi.org/10.1155/2021/9919333

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Abstract

Cluster growth models are utilized for a wide range of scientific and engineering applications, including modeling epidemics and the dynamics of liquid propagation in porous media. Invasion percolation is a stochastic branching process in which a network of sites is getting occupied that leads to the formation of clusters (group of interconnected, occupied sites). The occupation of sites is governed by their resistance distribution; the invasion annexes the sites with the least resistance. An iterative cluster growth model is considered for computing the expected size and perimeter of the growing cluster. A necessary ingredient of the model is the description of the mean perimeter as the function of the cluster size. We propose such a relationship for the site square lattice. The proposed model exhibits (by design) the expected phase transition of percolation models, i.e., it diverges at the percolation threshold p c . We describe an application for the porosimetry percolation model. The calculations of the cluster growth model compare well with simulation results.

Highlights

Percolation theory [1,2,3] has been developed to study the properties of connected clusters in graphs and their associated percolation processes. e simplest problem arising from percolation theory is the site/bond percolation: a regular lattice is considered in which either the cells or the edges are the relevant entities
We use the cluster growth model to predict the evolution of Φ and I for inputs (Φ0, I0) that correspond to the size and perimeter of one or more sides of a finite lattice. at is, for a 1-sided invasion of the site square lattice of size L, we have
We provided a detailed interface update formula for the site square lattice and extensively tested the cluster growth model with this lattice type

Summary

Introduction

Percolation theory [1,2,3] has been developed to study the properties of connected clusters in graphs and their associated percolation processes. e simplest problem arising from percolation theory is the site/bond percolation: a regular lattice is considered in which either the cells (sites) or the edges (bonds) are the relevant entities. The critical BRP is Complexity larger as each site has only a small group of local contacts; this is why reducing contacts (e.g., travel restrictions) are effective measures of defense Another way to stop the spread of the disease is to immunize people. Bak and Kalmar-Nagy introduced the porosimetry percolation model [20, 21], which is a variant of the classical invasion percolation model, to capture the dynamics of intrusion of nonwetting liquid into porous medium, such as in mercury injection porosimetry [22, 23] In this model, the occupation of the sites is controlled by an external field, the pressure p ∈ [0, 1] (analogous to the occupation probability of the classical percolation problem). We compare the cluster size evolutions obtained with both the cluster growth model and porosimetry percolation simulations.

Preliminaries

The Cluster Growth Model

Application for Porosimetry Percolation

Conclusions