Fractal Structure of Equipotential Curves on a Continuum Percolation Model

TL;DR

This paper numerically studies the fractal structure of equipotential curves in a 2D continuum percolation model, revealing how their fractal dimension varies with volume fraction and peaks at the percolation threshold.

Contribution

It introduces the concept of quasi-equipotential clusters with fractal structures and analyzes their fractal dimensions in relation to percolation properties.

Findings

Fractal dimension peaks at the percolation threshold.

Quasi-equipotential clusters exhibit step-like local values.

Fractal dimension ranges from 1.00 to 1.257.

Abstract

We numerically investigate the electric potential distribution over a two-dimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear the {\it{quasi-equipotential clusters}} which approximately and locally have the same values like steps and stairs. Since the quasi-equipotential clusters has the fractal structure, we compute the fractal dimension of equipotential curves and its dependence on the volume fraction over $[0,1]$. The fractal dimension in [1.00, 1.257] has a peak at the percolation threshold $p_c$.