论文标题
通过对数确定的协方差矩阵和液晶的应用
Quasi-entropy by log-determinant covariance matrix and application to liquid crystals
论文作者
论文摘要
使用协方差矩阵的对数确定器在$ so（3）$上通过密度函数平均的张量构建了准内层。它用作熵的替代，用于从涉及积分的约束最小化中得出的张量。准内拷贝是具有原始熵的基本特性的基本函数。它将协方差矩阵限制为正确定，严格凸出，并且在旋转下是不变的。此外，当通过对称性减小时，它会使对称性的消失张量为零。为轴向对称性以及四面体和八面体对称性提供了显式表达式。准内拷贝用于讨论几种系统中的相变。结果与使用原始熵一致。此外，还提出了一些新的结果。
A quasi-entropy is constructed for tensors averaged by a density function on $SO(3)$ using the log-determinant of a covariance matrix. It serves as a substitution of the entropy for tensors derived from a constrained minimization that involves integrals. The quasi-entropy is an elementary function that possesses the essential properties of the original entropy. It constrains the covariance matrix to be positive definite, is strictly convex, and is invariant under rotations. Moreover, when reduced by symmetries, it keeps the vanishing tensors of the symmetry zero. Explicit expressions are provided for axial symmetries up to four-fold, as well as tetrahedral and octahedral symmetries. The quasi-entropy is utilized to discuss phase transitions in several systems. The results are consistent with using the original entropy. Besides, some novel results are presented.
