Dhar Deepak, Rajesh R
Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, India.
The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India.
Phys Rev E. 2021 Apr;103(4-1):042130. doi: 10.1103/PhysRevE.103.042130.
We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×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×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}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≥2.
我们确定了在(k)很大的极限情况下，由尺寸为(k×1)和(1×k)的棒对(L×M)方形晶格进行完全覆盖时熵的渐近行为。我们表明，只有当(L)和(M)中至少有一个是(k)的倍数时，完全覆盖才有可能，并且所有允许的构型都可以从所有棒都平行的标准构型出发，仅使用基本的翻转操作来达到，这些操作是将(k×k)的平行水平棒的正方形替换为垂直棒，反之亦然。在(k)很大的极限情况下，我们表明每个位点的熵(S_{2}(k))趋于(Ak^{-2}\ln k)，其中(A = 1)。基于微扰级数展开，我们推测每个位点熵的这种大(k)行为是超普适的，并且在所有(d\geq2)的(d)维超立方晶格上都继续成立。
