Entropy of full covering of the kagome lattice by straight trimers

TL;DR

This paper calculates the entropy of covering a kagome lattice with straight trimers by establishing a correspondence with dimer coverings on a hexagonal lattice, providing an exact entropy value.

Contribution

It introduces a novel 2-to-1 mapping between trimer coverings on the kagome lattice and dimer coverings on a related hexagonal lattice, enabling exact entropy calculation.

Findings

Entropy per trimer is approximately 0.3231.

Established a precise mathematical relationship between kagome trimers and hexagonal dimers.

Provided an explicit integral formula for the entropy value.

Abstract

We consider the number of ways all the sites of a kagome lattice can be covered by non-overlapping linear rigid rods where each rod covers 3 sites. We establish a 2-to-1 correspondence between the configurations of trimers on the kagome lattice to the covering by dimers of a related hexagonal lattice to show that entropy of coverings per trimer $s_{\text{tri,kag}}$ equals the entropy per dimer $ s_{\text{dim,hex}} $, and is given by $ s_{\text{tri,kag}} = s_{\text{dim,hex}} = \frac{1}{2 \pi} \int_0^{ 2 \pi/3} \log( 2 + 2 \cos k) dk \approx 0.323065947\ldots$.