刚性分子形成的局部各向异性分类：对称性和张量
Classifying local anisotropy formed by rigid molecules: symmetries and tensors
摘要：
本文考虑一个无穷小体积，其中有许多刚性的同类分子，并讨论了在这种体积的局部各向异性的描述和分类张量。首先，我们研究了刚性分子的对称性，这是由一个点群在$SO(3)$。对于$SO(3)$中的每个点群，我们发现张量在群中的旋转下是不变的。这些张量应该是对称的和无迹的。我们写下显式表达式。然后选择描述局部各向异性的序参量作为关于密度函数平均的一些不变张量。其次，我们讨论了局部各向异性的分类由整个无穷小体积的对称性。这种介观对称性可以通过最大熵态意义下的序参量张量来识别。对于包含不同分子对称性的序参量张量集合，我们给出了介观对称性的分类，并考察了三重、四重和多面体对称性。
英文摘要：
We consider an infinitesimal volume where there are many rigid molecules of the same kind, and discuss the description and classification of the local anisotropy in this volume by tensors. First, we examine the symmetry of a rigid molecule, which is described by a point group in $SO(3)$. For each point group in $SO(3)$, we find the tensors invariant under the rotations in the group. These tensors shall be symmetric and traceless. We write down the explicit expressions. The order parameters to describe the local anisotropy are then chosen as some of the invariant tensors averaged about the density function. Next, we discuss the classification of local anisotropy by the symmetry of the whole infinitesimal volume. This mesoscopic symmetry can be recognized by the value of the order parameter tensors in the sense of maximum entropy state. For some sets of order parameter tensors involving different molecular symmetries, we give the classification of mesoscopic symmetries, in which the three-fold, four-fold and polyhedral symmetries are examined.
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