Entropy of fully packed hard rigid rods on d-dimensional hypercubic lattices

We determine the asymptotic behavior of the entropy of full coverings of a L x M square lattice by rods of size k x 1 and 1 x 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 x 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 Ak(-2) ln k, 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 >= 2.

Automated summary

The study investigates the asymptotic behavior of entropy when fully covering a square lattice with rods of specific sizes in the limit of large k. The research reveals the conditions under which full coverage is possible and the basic flip moves between configurations. In the large k limit, per-site entropy tends towards a specific mathematical function.