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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|ISOTROPIC PHASE OF LIQUID CRYSTALS

Above $T_{c}$, liquid crystals lose their directional order and behave in many respects like liquids. All bulk physical parameters also assume an isotropic form, although the molecules are anisotropic.

The isotropic phase is, nevertheless, a very interesting and important phase for both fundamental and applied studies. It is fundamentally interesting because of the existence of short-range order, which gives rise to the critical temperature dependence of various physical parameters just above the phase transition temperature. These critical behaviors provide a good testing ground for liquid crystal physics.

On the other hand, recent studies have also shown that isotropic liquid crystals may be superior in many ways for constructing practical nonlinear optical devices (see Chapter 12), in comparison to the other liquid crystalline phases (see Chapter 8). In general, the scattering loss is less and thus allows longer interaction lengths, and relaxation times are on a much faster scale. These properties easily make up for the smaller optical nonlinearity for practical applications.

## 物理代写|光学代写Optics代考|Free Energy and Phase Transition

We begin our discussion of the isotropic phase of liquid crystals with the free energy of the system, following deGennes’ pioneering theoretical development $[1,2]$. The starting point is the order parameter, which we denote by $Q$.

In the absence of an external field, the isotropic phase is characterized by $Q=0$; the minimum of the free energy also corresponds to $Q=0$. This means that, in the Landau expansion of the free energy in terms of the order parameter $Q$, there is no linear term in $Q$; that is,
$$F=F_{0}+\frac{1}{2} A(T) \sum_{\rho, \alpha} Q_{\alpha \beta} Q_{\beta \alpha} \frac{1}{3} B(T) \sum_{\alpha, \beta, \gamma} Q_{\alpha \beta} Q_{\alpha \gamma} Q_{\gamma \alpha}+O(Q 4)$$
where $F_{0}$ is a constant and $A(T)$ and $B(T)$ are temperature-dependent expansion coefficients:
$$A(T)=\alpha\left(T-T^{}\right)$$ where $T^{}$ is very close to, but lower than, $T_{c}$. Typically, $T_{c}-T_{c}^{*}=1 \mathrm{~K}$.
Note that $F$ contains a nonzero term of order $Q[3]$. This odd function of $Q$ ensures that states with some nonvanishing value of $Q$ (e.g. due to some alignment of molecules) will have different free-energy values depending on the direction of the alignment. For example, the free energy for a state with an order parameter $Q$ of the form
$$Q_{1}=\left(\begin{array}{ccc} -\xi & 0 & 0 \ 0 & -\xi & 0 \ 0 & 0 & 2 \xi \end{array}\right)$$

# 光学代考

## 物理代写|光学代写Optics代考|Free Energy and Phase Transition

$$F=F_{0}+\frac{1}{2} A(T) \sum_{\rho, \alpha} Q_{\alpha \beta} Q_{\beta \alpha} \frac{1}{3} B(T) \sum_{\alpha, \beta, \gamma} Q_{\alpha \beta} Q_{\alpha \gamma} Q_{\gamma \alpha}+O(Q 4)$$

$$A(T)=\alpha(T-T)$$

$$Q_{1}=\left(\begin{array}{lllllllll} -\xi & 0 & 0 & 0 & -\xi & 0 & 0 & 0 & 2 \xi \end{array}\right)$$

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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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