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

TL;DR

This paper introduces a quasi-entropy based on the log-determinant of a covariance matrix for tensors on SO(3), useful for analyzing phase transitions in liquid crystals, maintaining key properties of the original entropy.

Contribution

It develops a new quasi-entropy function for tensors that preserves essential entropy properties and applies it to study phase transitions in liquid crystal systems.

Findings

Quasi-entropy constrains covariance matrices to be positive definite.

It is strictly convex and rotation-invariant.

Results align with traditional entropy methods, with some novel insights.

Abstract

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.