Entropy of fully packed hard rigid rods on $d$-dimensional hyper-cubic lattices

Abstract

We determine the asymptotic behavior of the entropy of full coverings of a $L\times M$ square lattice by rods of size $k\times 1$ and $1\times k$, in the limit of large $k$. We show that full coverage is possible only if at least one of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a $k\times k$ square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large $k$, we show that the entropy per site $S_2(k)$ tends to $A{k}^{-2}\ln k$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is super-universal and continues to hold on all $d$-dimensional hyper-cubic lattices, with $d\ge 2$.

arXiv comment13 pages, 12 figures

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Cited by in corpus (6)

• The phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice
• Entropy of rigid k-mers on a square lattice
• Entropy of different phases formed by soft rods
• Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices
• Phases of the hard-plate lattice gas on a three-dimensional cubic lattice
• Entropy of fully-packed rigid rods on generalized Husimi trees: a route to the square lattice limit