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

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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|Lattice model

An analytically simple but useful variation of the coarse grained description given above is the lattice model. In the model the space continuum is discretized into $M=V / v_{0}$ lattice sites (Fig. 4.7). Each lattice site can be empty or occupied by a particle, so that this is a two-state model with each site characterized by occupation number $n_{i}=0$ or 1 . This model incorporates the excluded volume effects. Once we have $M$ lattice sites onto which a particle can bind, we then have $M-1$ sites for the next particle, so on. The number of ways of configuring $N$ particles in $\boldsymbol{M}$ distinct lattice sites is $M(M-1) \ldots(M-N-1)$. With the factor $1 / N !$ multiplied due to indistinguishability of $N$ particles, the partition function is given as
$$Z_{U}=\frac{1}{N !} M(M-1) \ldots(M-N-1),$$
which is equal to
$$Z_{U}=\frac{M !}{N !(M-N) !}$$
This is the same as the factor which we already considered when two state model was first introduced in chap. 3. If $N \ll M, M ! /(M-N) ! \approx M^{N}=\left(V / v_{0}\right)^{N}$ and $Z_{U}$ is reduced to $Z_{U}^{0}$ (4.85), the partition function of non-interacting particles.
P4.6 Show that the equation of state of the lattice gas is given as:
$$\frac{p V}{N k_{B} T}=-\frac{1}{\theta} \ln (1-\theta) .$$
where $0 \leq \theta=N / M \leq 1$. The chemical potential is given by
$$\mu=k_{B} T \ln \left(\frac{\theta}{1-\theta}\right),$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Mesoscopic Degrees of Freedom

The macroscopic behavior of an equilibrium at a fixed temperature $T$ is determined formally by the Hamiltonian, through the relation (3.32) supplemented by (3.35):
$$e^{-\beta F}=\sum_{\mathcal{M}} e^{-\beta \mathcal{H}{\mathcal{M}}},$$
where $F$ is the Helmholtz free energy, $\sum_{\mathcal{M}}(\cdot)$ denotes the summation over all microscopic degrees of freedom represented by $\mathcal{M}$; if $\mathcal{M}$ is continuous, the summation signifies the integration, e.g. for a classical particle system the integration over the phase space spanned by all the particles. For illustrative purpose we consider a combined system (solution) of solute and solvent at $T$, including the solution of polymer chains where the monomers are linearly connected solute particles. The microscopic states are given by the phase space of all solute molecules and solvent molecules, and, if required, the microscopic quantum states that underlie the molecules. Considering the appreciable complexity seen even in the statistical mechanics of simple fluids (Chap. 4), an attempt to conduct the standard scheme using the formalism, (5.1) would be very costly. Even if we could do so the results can obscure the most salient and interesting features of the system.

We may wisely abandon the full microscopic description and choose a coarse-grained description in terms of the relevant degrees of freedom, represented by $Q$, in terms of which we have
$$e^{-\beta F}=\sum_{\mathcal{Q}} e^{-\beta \mathcal{F}{\mathcal{Q}}}$$

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

$$Z_{U}=\frac{1}{N !} M(M-1) \ldots(M-N-1)$$

$$Z_{U}=\frac{M !}{N !(M-N) !}$$

$P 4.6$ 证明晶格气体的状态方程为:
$$\frac{p V}{N k_{B} T}=-\frac{1}{\theta} \ln (1-\theta)$$

$$\mu=k_{B} T \ln \left(\frac{\theta}{1-\theta}\right)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Mesoscopic Degrees of Freedom

$$e^{-\beta F}=\sum_{\mathcal{M}} e^{-\beta \mathcal{H M}},$$

$$e^{-\beta F}=\sum_{\mathcal{Q}} e^{-\beta \mathcal{F} \mathcal{Q}}$$

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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。