SIAM Journal on Applied Mathematics, Volume 80, Issue 6, Page 2518-2546, January 2020.
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 threefold, fourfold, and polyhedral symmetries are examined.

中文翻译：

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对刚性分子形成的局部各向异性进行分类：对称性和张量

SIAM应用数学杂志，第80卷，第6期，第2518-2546页，2020年1月。
我们考虑存在许多相同种类的刚性分子的无穷小体积，并通过张量讨论该体积中局部各向异性的描述和分类。首先，我们检查刚性分子的对称性，该对称性由$ SO（3）$中的点组描述。对于$ SO（3）$中的每个点组，我们发现该组旋转下的张量不变。这些张量应对称且无迹。我们写下显式表达式。然后选择描述局部各向异性的有序参数作为一些关于密度函数平均的不变张量。接下来，我们讨论通过整个无限小体积的对称性对局部各向异性的分类。这种介观对称性可以通过最大熵状态意义上的阶次参数张量的值来识别。