主题：Quasi-entropy and applications in liquid crystals 拟熵及在液晶中的应用

主讲人：中科院数学与系统科学研究院  徐劼 副研究员

主持人：yh533388银河 王韦龙

时间：2023年11月24日（周五）10:00-11:00

地点：柳林校区通博楼B412会议室

主办单位：yh533388银河

主讲人简介：

徐劼，中国科学院数学与系统科学研究院副研究员。徐劼博士在液晶建模与模拟、梯度流的数值方法方面取得了突出的研究成果，代表工作包括：针对复杂液晶分子发展了基于分子理论的系统建模方法；独立提出了拟熵使模型可计算化；提出了梯度流的标量辅助变量方法。

内容提要：

Liquid crystals are featured by local anisotropy usually described by angular moment tensors. The free energy of tensors needs a stabilizing entropy term. When non-axisymmetric molecules are involved, two classical approaches to write down an entropy term, quartic polynomial and maximum entropy state, both become too complicated. Meanwhile, the maximum entropy state, particularly in cases of non-axisymmetric molecules, seems the only reasonable way of closure approximation in dynamics, but it would make the model not computable.

We propose an elementary-function substitution of the original entropy (by maximum entropy state), called quasi-entropy, aiming to resolve the above problems. The quasi-entropy maintains the essential properties of the original entropy: strict convexity; positive-definiteness of covariance matrix; rotational invariance; consistency in symmetry reduction. Homogeneous phase diagrams of several representative cases match well with classical results. A novel closure approximation is also available. A preliminary application is deriving biaxial frame hydrodynamics from tensor model.

液晶具有局部各向异性的特点，通常用角矩张量来描述。张量的自由能需要一个稳定的熵项。当涉及非轴对称分子时，写熵项的两种经典方法，四次多项式和最大熵态，都变得过于复杂。同时，最大熵态，特别是在非轴对称分子的情况下，似乎是动力学中闭合近似的唯一合理方法，但它会使模型不可计算。

为了解决上述问题，我们提出了一种原始熵（由最大熵态）的初等函数替换，称为拟熵。拟熵保持了原始熵的本质性质：严格凸性；协方差矩阵的正定性；旋转不变性；对称性约简的一致性。几个代表性案例的齐次相图与经典结果吻合良好。一个新颖的封闭近似是有效的。一个初步的应用是从张量模型推导双轴框架流体力学。