Connectedness percolation of fractal liquids

TL;DR

This paper extends connectedness percolation theory to fractal liquids, showing how percolation thresholds vary with fractal dimensions and highlighting the limitations of current theoretical closures.

Contribution

It introduces a novel application of percolation theory to fractal dimensions and analyzes the effects of hard-core interactions on percolation thresholds.

Findings

Percolation threshold interpolates between integer dimensions.

Threshold decreases monotonically with increasing fractal dimension.

Theory suggests absence of threshold below two dimensions, due to closure breakdown.

Abstract

We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is only significant for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.