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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|光学代写Optics代考|Long- and Short-range Order

The order parameter, defined by Eq. (2.2) and its variants such as Eqs. (2.4) and (2.8), is an average over the whole system and therefore provides a measure of the long-range orientation order. The smaller the fluctuation of the molecular axis from the director axis orientation direction, the closer the magnitude of $S$ is to unity. In a perfectly aligned liquid crystal, as in other crystalline materials, $\left\langle\cos ^2 \theta\right\rangle=1$ and $S=1$; on the other hand, in a perfectly random system, such as ordinary liquids or the isotropic phase of liquid crystals, $\left\langle\cos ^2 \theta\right\rangle=\frac{1}{3}$ and $S=0$.

An important distinction between liquid crystals and ordinary anisotropic or isotropic liquids is that, in the isotropic phase, there could exist a so-called short-range order [1,2]; that is, molecules within a short distance of one another are correlated by intermolecular interactions [4]. These molecular interactions may be viewed as remnants of those existing in the nematic phase. Clearly, the closer the isotropic liquid crystal is to the phase transition temperature, the more pronounced the short-range order and its manifestations in many physical parameters will be. Short-range order in the isotropic phase gives rise to interesting critical behavior in the response of the liquid crystals to externally applied fields (electric, magnetic, and optical) (see Section 2.3.2).

As pointed out at the beginning of this chapter, the physical and optical properties of liquid crystals may be roughly classified into two types: one pertaining to the ordered phase, characterized by long-range order and crystalline-like physical properties; the other pertaining to the so-called disordered phase, where a short-range order exists. All these order parameters show critical dependences as the temperature approaches the phase transition temperature $T_c$ from the respective directions.

## 物理代写|光学代写Optics代考|MOLECULAR INTERACTIONS AND PHASE TRANSITIONS

In principle, if the electronic structure of a liquid crystal molecule is known, one can deduce the various thermodynamical properties. This is a monumental task in quantum statistical chemistry that has seldom, if ever, been attempted in a quantitative or conclusive way. There are some fairly reliable guidelines, usually obtained empirically, that relate molecular structures with the existence of the liquid crystal mesophases and. less reliably, the corresponding transition temperatures.

One simple observation is that to generate liquid crystals, one should use elongated molecules. This is best illustrated by the $n \mathrm{CB}$ homolog $[5](n=1,2,3, \ldots)$. For $n \leq 4$, the material does not exhibit a nematic phase. For $n=5-7$, the material possesses a nematic range. For $n>8$, smectic phases begin to appear.

Another reliable observation is that the nematic $\rightarrow$ isotropic phase transition temperature $T_c$ is a good indicator of the thermal stability of the nematic phase [6]; the higher the $T_c$, the greater is the thermal stability of the nematic phase. In this respect, the types of chemical groups used as substituents in the terminal groups or side chain play a significant role – an increase in the polarizability of the substituent tends to be accompanied by an increase in $T_c$.

Such molecular-structure-based approaches are clearly extremely complex and often tend to yield contradictory predictions because of the wide variation in the molecular electronic structures and intermolecular interactions present. In order to explain the phase transition and the behavior of the order parameter in the vicinity of the phase transition temperature, some simpler physical models have been employed [6]. For the nematic phase, a simple but quite successful approach was introduced by Maier and Saupe [7]. The liquid crystal molecules are treated as rigid rods, which are correlated (described by a long-range order parameter) with one another by Coulomb interactions. For the isotropic phase, deGennes introduced a Landau type of phase transition theory [1-3], which is based on a short-range order parameter.

The theoretical formalism for describing the nematic $\rightarrow$ isotropic phase transition and some of the results and consequences are given in the next section. This is followed by a summary of some of the basic concepts introduced for the isotropic phase.

# 光学代考

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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