# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYSICS334

#### Doug I. Jones

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• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
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## 物理代写|统计物理代写Statistical Physics of Matter代考|Statistical Mechanics of Fluids and Solutions

Biological components function often in watery environments. Biological fluids are either water solvent or various aqueous solutions and suspensions of ions and macromolecules, with which virtually all chapters of this book are concerned. In this chapter we start with a review of how the canonical ensemble method of statistical mechanics can be used to derive some basic properties of simple, classical fluids that consist of small molecules. We derive the well-known thermodynamic properties of non-interacting gases either in the absence or in the presence of external forces. For dilute and non-dilute fluids, we study how the inter-particle interactions give rise to the spatial correlations in the fluids, which affects the thermodynamic behaviors.

These results, which are essential for a simple fluid for its own, can be extended to aqueous solutions of colloids and macromolecules; e.g., the results of dilute simple gas can be directly applied to dilute solutions. We outline coarse-grained descriptions in which the solutions are treated as the fluids of solutes undergoing the solvent-averaged effective interactions. As a particularly simple but useful variation we shall introduce the lattice model.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Phase-Space Description of Fluids

Consider a simple fluid consisting of $N$ identical classical particles of mass $m$ each with no internal degrees of freedom. The fluid is confined in a rectangular volume $V$ with sides $L_{x}, L_{y}, L_{z}$ and kept at a temperature $T$. For a classical but microscopic description, the microstate $\mathcal{M}$ of the system is specified by a point in $\mathbf{6} N$ dimensional phase space $\Gamma=\left(p_{1}, r_{1}, \ldots p_{i}, r_{i}, \ldots p_{N}, r_{N}\right) \equiv\left{p_{i}, r_{i}\right}$ where $p_{i}, r_{i}$ are the three-dimensional momentum and position vectors of the $i$-th particles. The particles are in motion with the Hamiltonian
$$\mathcal{H}\left{\boldsymbol{p}{i}, \boldsymbol{r}{i}\right}=K\left{\boldsymbol{p}{i}\right}+U\left{\boldsymbol{r}{i}\right}+\Phi\left{\boldsymbol{r}{i}\right} .$$ Here $K\left{\boldsymbol{p}{i}\right}=\sum_{i=1}^{N} \boldsymbol{p}{i}^{2} /(2 m)$ is net kinetic energy of the system, $U\left{\boldsymbol{r}{i}\right}=$ $\sum_{i=1}^{N} u\left(\boldsymbol{r}{i}\right)$ is the net external potential energy, where $u\left(\boldsymbol{r}{i}\right)$ is one body potential energy of particle $i . \Phi\left{\boldsymbol{r}{i}\right}=\sum{i>j} \varphi\left(\boldsymbol{r}{i}-\boldsymbol{r}{j}\right)$, the net interaction potential energy, which is the sum of $N(N-1) / 2$ pairwise interaction potential energies between particles positioned at $\boldsymbol{r}{i}$ and $\boldsymbol{r}{j}, \varphi\left(\boldsymbol{r}{i}-\boldsymbol{r}{j}\right) \equiv \varphi\left(\boldsymbol{r}_{i j}\right)$.

The canonical microstate distribution (3.31) for this system is the $N$ particle phase-space distribution function:
$$P\left{\boldsymbol{p}{i}, \boldsymbol{r}{i}\right}=\frac{1}{N !} \frac{1}{h^{3 N}} e^{-\beta \mathcal{H}\left{\boldsymbol{p}{i}, \boldsymbol{r}{i}\right}} / Z_{N}$$
This is the joint probability distribution with which the $N$ particles have their all positions and momenta at $\boldsymbol{p}{1}, \boldsymbol{r}{1} ; \ldots \boldsymbol{p}{i}, \boldsymbol{r}{i} ; \ldots \boldsymbol{p}{N}, \boldsymbol{r}{N}$ simultaneously. The partition function $Z_{N}$ is given as the $6 N$-dimensional integral:
\begin{aligned} Z_{N} &=\frac{1}{N !} \frac{1}{h^{3 N}} \int d \boldsymbol{\Gamma} e^{-\beta \mathcal{H}(\boldsymbol{\Gamma})} \ &=\frac{1}{N !} \frac{1}{h^{3 N}} \int \ldots \int d \boldsymbol{p}{1} d \boldsymbol{r}{1} \ldots d \boldsymbol{p}{N} d \boldsymbol{r}{N} e^{-\beta \mathcal{H}\left{\boldsymbol{p}{i}, r{i}\right}} \end{aligned}

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Phase-Space Description of Fluids

，净相互作用势能，它是 $N(N-1) / 2$ 位于的粒子之间的成对相互作用势能 $r i$ 和
$\boldsymbol{r} j, \varphi(\boldsymbol{r} i-\boldsymbol{r} j) \equiv \varphi\left(\boldsymbol{r}{i j}\right)$ 该系统的典型微状态分布 (3.31) 是 $N$ 粒子相空间分布函数: 这是联合概率分布 $N$ 粒子的所有位置和动量都在 $\boldsymbol{p} 1, \boldsymbol{r} 1 ; \ldots \boldsymbol{p} i, \boldsymbol{r} i ; \ldots \boldsymbol{p} N, \boldsymbol{r} N$ 同时。分 区函数 $Z{N}$ 给出为 $6 N$ 维积分.

## 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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