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## 物理代写|统计物理代写Statistical Mechanics代考|Potts Model

The Potts model $[144,145]$ was introduced by Renfrey B. Potts as a generalization of the Ising model with $q \geq 2$ states. It is a very versatile model due to its applications in many fields including sociology, biology and material science. The Hamiltonian of the Potts model is
$$\mathcal{H}({\sigma})=-J \sum_{\langle i, j\rangle} \delta_{\sigma_i \sigma_j}-H \sum_i \sigma_i,$$
where $\sigma_i \in{1, \ldots, q}$ and $\delta_{\sigma_i \sigma_j}$ is unity when nodes $i$ and $j$ are in the same state (Kronecker delta). The Potts model exhibits a first order transition at the critical temperature in two dimensions for $q>4$, and for $q>2$ for dimensions larger than three. ${ }^1$ For $q=2$, the Potts model is equivalent to the Ising model. Moreover, there exists a connection between the Potts model and bond percolation. Kasteleyn (see Figure 6.1) and Fortuin demonstrated that the two models have related partition functions [147]. A thermodynamic system is characterized by its partition function from which all thermodynamic quantities can be derived. Therefore, the partition function completely describes the thermodynamic properties of a system. If two systems have the same partition function (up to a multiplicative constant), we consider these two systems as equivalent. We will derive the relation between the Potts model and bond percolation in the following section.

## 物理代写|统计物理代写Statistical Mechanics代考|The Kasteleyn and Fortuin Theorem

We consider the Potts model not just on a square lattice but on an arbitrary graph that consists of nodes and bonds $v$. Each node can be in $q$ possible states and each connection leads to an energy cost of unity if two connected nodes are in a different state and of zero if they are in the same state:
$$E=J \sum_v \epsilon_v \text { with } \epsilon_v= \begin{cases}0, & \text { if endpoints are in the same state, } \ 1, & \text { otherwise. }\end{cases}$$
As we show in Figure 6.2, we introduce contraction and deletion operations of a bond $v$ on the graph. As shown in detail in the gray box on the next page, the partition function of the Potts model can be split up into two partition functions, one for the deleted and one for the contracted graph. After applying these operations to every bond, the graph is reduced to a set of separated points corresponding to clusters of nodes that are connected and in the same state out of $q$ states. The partition function of the Potts model $Z=\sum_X e^{-\beta E(X)}$ reduces to
$$Z=\sum_{\begin{array}{c} \text { configurations of } \ \text { bond percolation } \end{array}} q^{# \text { of clusters }} p^c(1-p)^d=\left\langle q^{# \text { of clusters }}\right\rangle_{\mathrm{b}},$$
where $c$ and $d$ are, respectively, the numbers of contracted and deleted bonds and $p=1-\exp (-\beta J)$. In the limit $q \rightarrow 1$, one can derive the generating function of bond percolation. In bond percolation, an edge of a graph is occupied with probability $p$ and vacant with probability $1-p$. Equation (6.3) constitutes a fundamental relation between a purely geometrical model (bond percolation) and a magnetic model described by a Hamiltonian (Potts model) [148]. Interestingly, we can now choose noninteger values for $q$. This was meaningless in the original definition of the Potts model, because $q$ was introduced as the state of each site.

## 物理代写|统计物理代写|统计力学代考|Potts模型

$$\mathcal{H}(sigma)=-J\sum_{langle i, j\rangle}。\δ{sigma_i sigma_j}-Hsum_i `sigma_i$$ 其中$sigma_i 处于1, \ldots, q$和$delta{sigma_i \sigma_j}$在节点$i$和$j$处于同一状态时为统一（Kronecker Delta）。对于$q>4$，Potts模型在二维的临界温度下表现出一阶转换，对于大于三维的$q>2$，表现出一阶转换。${ }^1$ 对于$q=2$，Potts模型等同于Ising模型。此外，在Potts模型和债券渗流之间存在着一种联系。Kasteleyn（见图6.1）和Fortuin证明这两个模型有相关的分区函数[147]。一个热力学系统的特征是它的分区函数，所有的热力学量都可以从分区函数中得到。因此，分区函数完全描述了一个系统的热力学特性。如果两个系统具有相同的分区函数（最多是一个乘法常数），我们认为这两个系统是等价的。我们将在下一节中推导出Potts模型和债券渗流之间的关系。

## 物理代写|统计物理代写|统计力学代考|Kasteleyn和Fortuin定理

$E=J \sum_v \epsilon_v$，$epsilon_v={0, \quad$ 如果端点处于同一状态，1, \quad$否则。 正如我们在图6.2中所示，我们在图上引入了键$v$的收缩和删除操作。正如下一页灰框中所详细显示的，Potts模型的分区函数可以分成两个分区函数，一个用于删除的图形，一个用于收缩的图形。将这些操作应用于每一个纽带后，图被还原为一组分离点，对应于$q$状态中的连接和相同状态的节点群。波茨模型的分区函数$Z=sum_X e^{-beta E(X)}$降低为 你不能在数学模式下使用’宏参数字符#’。 其中$c$和$d$分别是收缩键和删除键的数量，$p=1-exp (-\beta J)$。在极限$q\rightarrow 1$中，我们可以推导出债券渗滤的生成函数。在键合渗滤中，图的一条边被占用的概率为p$，空出的概率为1-p$。方程（6.3）构成了纯几何模型（键合渗滤）和由哈密顿模型（Potts模型）描述的磁性模型之间的一个基本关系[148]。有趣的是，我们现在可以为$q$选择非整数值。这在Potts模型的原始定义中是没有意义的，因为$q\$是作为每个部位的状态引入的。

## 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。