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

In the first part of this chapter, our goal is to provide an overview of methods to computationally study systems that can be described within the framework of equilibrium statistical mechanics. We therefore first introduce the most important concepts from classical statistical mechanics that enable us to mathematically capture the macroscopic properties of interacting microscopic units. Historically, important contributions to the microscopic formulation of thermodynamics were made by Boltzmann and Gibbs $[105,106]$. In particular, Gibbs’s notion of a statistical ensemble enables us to interpret macroscopic physical quantities as averages over a large number of different configurations of a system’s interacting units.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

Let us consider a three-dimensional classical physical system that consists of $N$ particles. We need a three-component vector to describe the position of a certain particle and another three-component vector to describe its momentum. Therefore, we denote the canonical coordinates and corresponding conjugate momenta of all $N$ particles by $q_{1}, \ldots, q_{3 N}$ and $p_{1}, \ldots, p_{3 N}$, respectively. The $6 N$-dimensional space $\Gamma$ that results from the set of canonical coordinates defines the phase space. This concept has been introduced by Ludwig Boltzmann (see Figure 3.1). The considered $N$ particles could simply be uncoupled harmonic oscillators. In this case, the phase space of each single particle would look like the one we show in Figure 3.2. By keeping certain external parameters such as temperature and pressure constant, we can measure a macroscopic physical quantity by computing the time average
$$\langle Q\rangle_{T}=\lim {T \rightarrow \infty} \frac{1}{T} \int{0}^{T} Q(p(t), q(t)) \mathrm{d} t$$
over different realizations of the underlying microscopic states. However, computing the time average of a certain macroscopic quantity makes it necessary to determine the time evolution of all microscopic states. If the number of involved particles is large, this approach would be computationally infeasible. Instead of computing time averages, another possibility is to consider an average over an ensemble of systems in different microstates (i. e., specific microscopic configurations of a certain system) under the same macroscopic conditions.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

$$\langle Q\rangle_{T}=\lim T \rightarrow \infty \frac{1}{T} \int 0^{T} Q(p(t), q(t)) \mathrm{d} t$$

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

Networks can be characterized by their degree distribution $P(k)$. For a square lattice, the degree distribution is $P(k)=\delta_{k 4}$ (i.e., each node has degree $k=4$ ). If the distribution follows a power law
$$P(k) \propto k^{-\gamma},$$
the network is called scale-free [91]. Examples of networks that are approximately scale-free are the Internet and some social networks [92, 93]. One mechanism that produces scale-free networks is preferential attachment where each new node is more likely to attach to existing nodes with high degree. Mathematically, each new node will be attached to $m \leq m_{0}$ existing nodes and the attachment probability is proportional to the number of edges of the existing nodes. Here, $m_{0}$ is the initial number of nodes. Networks that result from this type of preferential attachment are called BarabásiAlbert networks and their degree distribution is $P(k) \propto k^{-3}$. We show an example of a Barabási-Albert network in Figure 2.44.

Networks that are generated by a random process are also called random-graph models [94, 95]. Another example of a random-graph model are Erdős-Rényi networks [96]. To generate them, we start with $N$ isolated nodes and add new edges between two uniformly at random selected nodes with probability $p$. The resulting degree distribution is binomial:
$$P(k)=\left(\begin{array}{c} N-1 \ k \end{array}\right) p^{k}(1-p)^{N-1-k},$$
where $N$ is the number of nodes.
Watts-Strogatz networks [97] also belong to the class of random-graph models and are generated by arranging nodes in a ring and connecting each node to its $K$ nearest neighbors. The $K / 2$ rightmost edges of each node are then rewired (i.e., reconnected) with probability $p$ (see Figure 2.45). The degree distribution of a Watts-Strogatz network is [98]
$$P(k)=\sum_{l=0}^{f(k, K)}\left(\begin{array}{c} K / 2 \ l \end{array}\right)(1-p)^{l} p^{K / 2-l} \frac{(p K / 2)^{k-K / 2-l}}{(k-K / 2-l) !} e^{-p K / 2}$$
where $f(k, K)=\min (k-K / 2, K / 2)$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

In the previous section, we defined the shortest path between two points on an arbitrary network $G(V, E)$, where $V$ is a set of nodes and $E$ is a set of edges. With the burning method (see Section 2.3.1), we have already seen an example of an algorithm that identifies the shortest-path length (“burning time”) on a percolation cluster. For a general graph $G(V, E)$, Edsger Dijkstra (see Figure 2.46) proposed a greedy algorithm ${ }^{7}$ in $1959 .$

Dijkstra’s Algorithm

1. We assign to every node a temporary distance value (zero for our initial node and infinity for all other nodes).
2. We set the initial node as “burned” and mark all other nodes as “unburned.” All unburned nodes are stored in a corresponding set.
3. For all unburned neighbors of the current node, we compute the corresponding distances. If the computed distance is smaller than the current one, we assign the smaller value. Otherwise, we keep the current value.
4. After having considered all of the neighbors of the current node, we mark the current node as “burned” and remove it from the set of unburned nodes. We will not consider a burned node again.
5. We stop the algorithm if we reach the destination node or if the smallest tentative distance among the nodes in the unvisited set is infinity.
6. Otherwise, we select the “unburned” node that is marked with the smallest distance, consider it as current node, and return to step $3 .$

When combined with additional optimization techniques, the run time of Dijkstra’s algorithm is proportional to $|E|+|V| \log |V|$. In addition to Dijkstra’s algorithm, there also exists the Bellman-Ford-Moore algorithm that can also handle negative weights [104].

## 物理代写|统计物理代与写Statistical Physics of Matter代考|Degree Distribution

$$P(k) \propto k^{-\gamma},$$

$$P(k)=(N-1 k) p^{k}(1-p)^{N-1-k},$$

Watts-Strogatz 网络 [97] 也属于随机图模型类，是通过将节点排列成一个环并将每个节点连接到它的 $K$ 最近的邻居。这 $K / 2$ 然后以概率重新连接 (即重新连接) 每个节点的最右边的边缬 $p$ (见图 2.45)。Watts-Strogatz网絡的度数分布为 [98]
$$P(k)=\sum_{l=0}^{f(k, K)}(K / 2 l)(1-p)^{l} p^{K / 2-l} \frac{(p K / 2)^{k-K / 2-l}}{(k-K / 2-l) !} e^{-p K / 2}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Dijkstra’s Algorithm

Dijkstra 算法

1. 我们为每个节点分配一个临时距离值（初始节点为零，所有其他节点为无穷大）。
2. 我们将初始节点设置为“已烧毁”，并将所有其他节点标记为“末烧录”。所有末销毁的节点都存储在相应的集合中。
3. 对于当前节点的所有末肤峣邻居，我们计算相应的距离。如果计算的距离小于当前距离，我们分配较小的值。否则，我们保 留当前值。 的节点。
4. 如果我们到达目标节点或者末访问集中的节点之间的最小暂定距离为无穷大，我们将停止算法。
当与其他优化技术结合使用时，Dijkstra 算法的运行时间与 $|E|+|V| \log |V|$. 除了 Dijkstra 算法之外，还有可以处理负权重的 Bellman-Ford-Moore 算法[104]。

## 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。

Posted on Categories:Statistical Physics of Matter, 物理代写, 统计物理

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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

Random walks are used to model many phenomena, including the motion of particles in liquids (Brownian motion, named after Robert Brown (see Figure 2.34)) [58, 59],

stock price fluctuations $[60,61]$,foraging behavior of animals [62], and

polymer chains at the theta point $[63,64]$.

Moreover, random walks are also important in search algorithms such as the famous PageRank algorithm [65]. We could continue this list with many more examples from various disciplines [66]. For the interested reader, we refer to Refs. [67, 68].

Mathematically, a random walk is a stochastic process

$$X_{t}=X_{0}+\sum_{j=1}^{t} Z_{j},$$

where $Z_{1}, Z_{2}, \ldots, Z_{t}$ denote $t$ independent random variables that can be scalars or vectors. We set $X_{0}=0$ and note that $X_{t}$ can be defined either on a lattice $\left(\mathbb{Z}^{d}\right)$ or on a continuous space $\mathbb{R}^{d}$. In one dimension, one possibility is to consider binary random variables $Z_{j}$. That is, $Z_{j}=1$ with probability $p$ and $Z_{j}=-1$ with probability $q=1-p$. Out of the $t$ total steps, the walker moves $l$ steps to the left $\left(Z_{j}=-1\right)$ and $m$ steps to the right $\left(Z_{j}=1\right)$. Thus, after $t$ steps, the walker is at position $X_{t}=m-l=2 m-t$. The probability that the walker is at position $X_{t}=2 m-t$ is distributed according to a binomial distribution [68]

$$P\left(X_{t}=2 m-t\right)=\left(\begin{array}{c} t \ m \end{array}\right) p^{m} q^{t-m} .$$

In Figure 2.35, we illustrate the binomial distribution of eq. (2.41) and the corresponding Gaussian approximation (see eq. (2.47)). We observe a data collapse when plotting $P(r, t) t^{1 / 2}$ against $r / t^{1 / 2}$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Self-Avoiding Walks

An important extension of a random walk is the concept of walks that cannot intersect their own trajectory (i.e., fulfill the condition of “excluded volume”). The most prominent example is the so-called self-avoiding walk (SAW). In one dimension, the walker would just move in one direction, and for dimensions $d \geq 4$, the intersection probability is vanishingly small, so we observe a regular random-walk behavior with fractal dimension $d_{\mathrm{f}}=2$ [68]. In two and three dimensions, the fractal dimensions are $d_{\mathrm{f}}=4 / 3$ and $d_{\mathrm{f}} \approx 5 / 3$, respectively [75-77]. The SAW was first introduced by Paul Flory (see Figure 2.39) to describe polymers in a good solvent $[75,78]$.

For the SAW, all configurations of same chain length $N$ have the same statistical weight [79]:
$$\Omega_{N}=\mu^{N} N^{\theta},$$

where $\mu$ denotes a chemical potential and $\theta=11 / 32$ for all two-dimensional lattices $[76,80]$. The generating function is equivalent to a grand canonical partition function (see Section 3.1.4)
$$Z(x)=\sum_{N} \Omega_{N} x^{N}=\sum_{N}(\mu x)^{N} N^{\theta},$$
where $x$ is the fugacity which corresponds to the statistical weight of adding one element to the chain. The radius of convergence of $Z(x)$ defines the critical fugacity $x_{c}=1 / \mu$. The average chain length is
$$\langle N\rangle=\left.\frac{\partial \ln (Z)}{\partial x}\right|{x=1}=\frac{\sum{N} N \Omega_{N}}{\sum_{N} \Omega_{N}}= \begin{cases}\text { finite, } & \text { if } x_{c}>1, \ \text { critical, } & \text { if } x_{c}=1, \ \text { infinite, } & \text { if } x_{c}<1\end{cases}$$

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

$\theta$ 点处的聚合物链 $[63,64]$.

$$X_{t}=X_{0}+\sum_{j=1}^{t} Z_{j},$$

$$P\left(X_{t}=2 m-t\right)=(t m) p^{m} q^{t-m} .$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Self-Avoiding Walks

$$\Omega_{N}=\mu^{N} N^{\theta},$$

$$Z(x)=\sum_{N} \Omega_{N} x^{N}=\sum_{N}(\mu x)^{N} N^{\theta},$$
$$\langle N\rangle=\frac{\partial \ln (Z)}{\partial x} \mid x=1=\frac{\sum N N \Omega_{N}}{\sum_{N} \Omega_{N}}=\left{\text { finite, } \quad \text { if } x_{c}>1, \text { critical, } \quad \text { if } x_{c}=1, \text { infinite }, \quad \text { if } x_{c}<1\right.$$

## 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。