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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|统计物理代写Statistical Physics of Matter代考|Nearly Straight Semiflexible Chains

In this section we shall study how a nearly-straight (rod-like) semiflexible chain thermally fluctuates. Such situations include a chain fragment that is shorter than the chain’s persistence length, or a chain of an arbitrary length that is stretched by a strong tension (Fig. 11.5), for which we seek the force-extension relation. Unlike the freely-jointed chain model for the flexible polymer situations (Chap. 3), the WLC calculation shown below is quite involved.

We express the segmental position at an arc length $s$ of the nearly-straight chain as
$$\boldsymbol{r}(s)=\boldsymbol{h}(s)+x(s) \hat{\boldsymbol{x}}$$
where $\hat{x}$ is the unit vector along the tension, and $\boldsymbol{h}(s)$ is the transverse undulation vector of small magnitude that varies slowly over the distance: $|\partial \boldsymbol{h} / \partial s| \ll 1$ (Fig. 11.6). A derivative of (11.19) with respect to $s$ is the unit tangent vector $\boldsymbol{u}(s)$, so we have $1=(\partial \boldsymbol{h} / \partial s)^2+(\partial x(s) / \partial s)^2$ or
$$\frac{\partial x(s)}{\partial s}=\left(1-\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2\right)^{1 / 2} \approx 1-\frac{1}{2}\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2 .$$
The extension of the chain along the force then is
$$\mathcal{X}=\int_0^L d s \frac{\partial x(s)}{\partial s}=L \quad 1 \int_0^L d s\left(\begin{array}{c} \partial \boldsymbol{h} \ \partial s \end{array}\right)^2,$$
of which the average is
$$X=L-\frac{1}{2} \int_0^L d s\left\langle\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)\right\rangle^2$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Force-Extension Relation

We want to find the transverse and longitudinal fluctuations of the chain and the relation between $X$ and the applied tension $f$ if present. To this end we need to first find $\left\langle(\partial \boldsymbol{h} / \partial s)^2\right\rangle$ from the effective Hamiltonian, which, for the general case with the $f$, reads

\begin{aligned} \mathcal{F} &=\frac{1}{2} \kappa \int_0^L\left(\frac{\partial^2 \boldsymbol{r}(s)}{\partial s^2}\right)^2 d s-f \mathcal{X} \ & \approx \frac{1}{2} \int_0^L\left[\kappa\left(\frac{\partial^2 \boldsymbol{h}}{\partial s^2}\right)^2+f\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2\right] d s \end{aligned}
the latter being in the “Gaussian level approximation” to correctly retain the second order in $\boldsymbol{h}$.

Because the functional form in real space, (11.23) is cumbersome to analyze, we introduce the Fourier transform:
\begin{aligned} &\boldsymbol{h}(q)=\int_0^L d s e^{-i q s} \boldsymbol{h}(s) \ &\boldsymbol{h}(s)=\frac{1}{L} \sum_q e^{i q s} \boldsymbol{h}(q) . \end{aligned}
We adopt the conventional periodic boundary condition, $\boldsymbol{h}(s)=\boldsymbol{h}(s+L)$, which allow $q$ to take $N$ discrete values
$$q_n=\frac{2 n \pi}{L}, \quad n=\pm 1, \pm 2 \ldots, \pm N / 2$$
where $N=L / a$ with $a$ being the microscopic length. For a very long chain $(N \gg 1)$, the choice of the boundary condition does not affect its bulk properties. However, for short chains, where the effects of the ends may not be negligible, one has to use the appropriate boundary condition that meets the actual situations.

# 统计物理代考

## 物理代写|统计物理代写物质统计物理代考|近直半柔性链

.

$$\boldsymbol{r}(s)=\boldsymbol{h}(s)+x(s) \hat{\boldsymbol{x}}$$

$$\frac{\partial x(s)}{\partial s}=\left(1-\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2\right)^{1 / 2} \approx 1-\frac{1}{2}\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2 .$$
，那么沿着力的链的延伸是
$$\mathcal{X}=\int_0^L d s \frac{\partial x(s)}{\partial s}=L \quad 1 \int_0^L d s\left(\begin{array}{c} \partial \boldsymbol{h} \ \partial s \end{array}\right)^2,$$
，其平均值是
$$X=L-\frac{1}{2} \int_0^L d s\left\langle\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)\right\rangle^2$$

## 物理代写|统计物理代写物质统计物理学代考|力与延伸的关系

. The Force-Extension – Relation

\begin{aligned} \mathcal{F} &=\frac{1}{2} \kappa \int_0^L\left(\frac{\partial^2 \boldsymbol{r}(s)}{\partial s^2}\right)^2 d s-f \mathcal{X} \ & \approx \frac{1}{2} \int_0^L\left[\kappa\left(\frac{\partial^2 \boldsymbol{h}}{\partial s^2}\right)^2+f\left(\frac{\partial \boldsymbol{h}}{\partial s}\right)^2\right] d s \end{aligned}

\begin{aligned} &\boldsymbol{h}(q)=\int_0^L d s e^{-i q s} \boldsymbol{h}(s) \ &\boldsymbol{h}(s)=\frac{1}{L} \sum_q e^{i q s} \boldsymbol{h}(q) . \end{aligned}

$$q_n=\frac{2 n \pi}{L}, \quad n=\pm 1, \pm 2 \ldots, \pm N / 2$$
，其中$N=L / a$和$a$是微观长度。对于一个很长的链$(N \gg 1)$，边界条件的选择并不影响它的大块性质。但对于短链，末端的影响可能不可忽略，必须使用符合实际情况的适当边界条件。

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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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