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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 asubstitution of the entropy for tensors derived from a constrained minimizationthat involves integrals. The quasi-entropy is an elementary function thatpossesses the essential properties of the original entropy. It constrains thecovariance matrix to be positive definite, is strictly convex, and is invariantunder rotations. Moreover, when reduced by symmetries, it keeps the vanishingtensors of the symmetry zero. Explicit expressions are provided for axialsymmetries 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 novelresults are presented.

Quasi-entropy by log-determinant covariance matrix and application to liquid crystals