d维晶格上相互作用的硬刚性转子的精确结果
Exact Results for Interacting Hard Rigid Rotors on a d-Dimensional Lattice
AI总结：
本文研究了d维晶格上枢轴点不对称的硬物体系统，证明在特定晶格间距范围内熵可精确由二聚体覆盖模型和第二维里系数决定，并通过数值模拟验证了取向分布函数等结果。
摘要：
我们研究了一组相同硬物体的熵，这些物体具有一般形状，每个物体以晶格间距为a的d维规则晶格的顶点为枢轴，但可以具有任意取向。当枢轴点不对称地位于物体上时，我们证明存在一个晶格间距a的范围，在该范围内，粒子在任何取向下最多只能与其一个邻居重叠。在此范围内，粒子系统的熵可以精确地用有限负活性下二聚体覆盖基础晶格的大配分函数来表示。该范围内的精确熵完全由第二维里系数决定。配分函数的计算也被证明可以约化为具有离散化取向的同一模型的计算。我们确定了某一位点取向概率分布函数的精确函数形式。这依赖于二聚体问题中给定活性下的二聚体密度，我们通过对相应的Mayer级数进行数值求和来确定该密度。这些结果通过数值模拟得到了验证。
英文摘要：
We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientation, a particle can overlap with at most one of its neighbors. In this range, the entropy of the system of particles can be expressed exactly in terms of the grand partition function of coverings of the base lattice by dimers at a finite negative activity. The exact entropy in this range is fully determined by the second virial coefficient. Calculation of the partition function is also shown to be reducible to that of the same model with discretized orientations. We determine the exact functional form of the probability distribution function of orientations at a site. This depends on the density of dimers for the given activity in the dimer problem, that we determine by summing the corresponding Mayer series numerically. These results are verified by numerical simulations.
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