Theory of continuum percolation II. Mean field theory

TL;DR

This paper develops a mean field theory for continuum percolation by mapping it to the Potts fluid, deriving critical exponents and density with a heuristic variational principle.

Contribution

It introduces a new mean field approach to continuum percolation using a mapping to the Potts fluid and a heuristic variational principle.

Findings

Critical exponents match mean field lattice percolation: β=1, γ=1, ν=0.5.

Critical density is given by ρ_c=1/ε, with ε involving the binding probability and interaction potential.

The theory aligns with known mean field exponents and provides a formula for critical density.

Abstract

I use a previously introduced mapping between the continuum percolation model and the Potts fluid to derive a mean field theory of continuum percolation systems. This is done by introducing a new variational principle, the basis of which has to be taken, for now, as heuristic. The critical exponents obtained are $\beta= 1$, $\gamma= 1$ and $\nu = 0.5$, which are identical with the mean field exponents of lattice percolation. The critical density in this approximation is $\rho_c = 1/\ve$ where $\ve = \int d \x \, p(\x) \{ \exp [- v(\x)/kT] - 1 \}$. $p(\x)$ is the binding probability of two particles separated by $\x$ and $v(\x)$ is their interaction potential.