Redefinition of site percolation in light of entropy and the second law of thermodynamics

TL;DR

This paper reexamines site and bond percolation on square lattices, highlighting issues with traditional entropy measures and proposing a new site percolation definition to align with thermodynamic principles.

Contribution

It introduces a novel definition of site percolation based on bond connectivity, resolving inconsistencies with entropy behavior and thermodynamic laws.

Findings

Traditional site percolation shows zero entropy at p=0, indicating simultaneous order and disorder.

Entropy with 1-p initially increases then decreases, contradicting thermodynamic expectations.

The new site percolation definition aligns entropy behavior with the second law of thermodynamics.

Abstract

In this article, we revisit random site and bond percolation in square lattice focusing primarily on the behavior of entropy and order parameter. In the case of traditional site percolation, we find that both the quantities are zero at $p=0$ revealing that the system is in the perfectly ordered and in the disordered state at the same time. Moreover, we find that entropy with $1-p$, which is the equivalent counterpart of temperature, first increases and then decreases again but we know that entropy with temperature cannot decrease. However, bond percolation does not suffer from either of these two problems. To overcome this we propose a new definition for site percolation where we occupy sites to connect bonds and we measure cluster size by the number of bonds connected by occupied sites. This resolves all the problems without affecting any of the existing known results.