Entropy of fully packed hard rigid rods on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-dimensional hypercubic lattices

Abstract

We determine the asymptotic behavior of the entropy of full coverings of a $L\ifmmode\times\else\texttimes\fi{}M$ square lattice by rods of size $k\ifmmode\times\else\texttimes\fi{}1$ and $1\ifmmode\times\else\texttimes\fi{}k$, 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 $k\ifmmode\times\else\texttimes\fi{}k$ 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 $A{k}^{\ensuremath{-}2}lnk$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is superuniversal and continues to hold on all $d$-dimensional hypercubic lattices, with $d\ensuremath{\ge}2$.

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