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## 物理代写|统计物理代写Statistical Physics of Matter代考|Microstates and Entropy

A macrostate of a macroscopic system at equilibrium is described by a few thermodynamic state variables. We consider here an isolated system with specified macrovariables, namely its internal energy $E$, its number of particles $N$, and generalized displacement $X_{i}$ such as its volume (see Table $2.1$ for the definitions). The number $N$ is usually very large (for the system consisting of one mole of gas, the number of molecules $N$ is the Avogadro number $N_{A}=6.022 \times 10^{23}$ ), and is often taken to be infinity (thermodynamic limit) in macroscopic systems. Many different microstates underlie a given macrostate. The set of microstates under a macrostate specified by these variables $\left(E, N, X_{i}\right)$ constitutes the microcanonical ensemble. For illustration, consider a one-mole classical gas that is isolated with its net energy $E$ and enclosing volume $V$. Microscopic states of the classical gas are specified by the positions and momenta of all $N$ particles. There are huge (virtually infinite) number of ways (microstates) that the particles can assume their positions and momenta without changing the values of $E, N, V$ of the macrostate. Each of these huge number of microstates constitutes a member of the microcanonical ensemble.

Suppose that the number of microstates (also called the multiplicity) belonging to this ensemble is $W\left(E, N, X_{i}\right)$. Then the central postulate of statistical mechanics is that each microstate $\mathcal{M}$ within this ensemble is equally probable:
$$P{\mathcal{M}}=\frac{1}{W\left(E, N, X_{i}\right)}$$
This equal-a-priori probability is the least-biased estimate under the constraints of fixed total energy. This very plausible postulate is associated with another fundamental equation that relates the macroscopic properties with the microscopic information, the so-called Boltzmann formula for entropy:
$$S\left(E, N, X_{i}\right)=k_{B} \ln W\left(E, N, X_{i}\right) .$$
where $k_{B}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}=1.38 \times 10^{-18} \mathrm{erg} / \mathrm{K}$ is the Boltzmann constant. Equation (3.2) is the famous equation inscribed on Boltzmann’s gravestone in Vienna, (Fig. 3.1) and is regarded as the cornerstone of statistical mechanics. It proclaims that the entropy is a measure of disorder; $S=0$ at the most ordered state where only one microstate is accessible $(W=1)$; the irreversible approach to an entropy maximum is due to emergence of most numerous microstates, i.e., most disordered state, which is attained at equilibrium. Furthermore it tells us that once $W$ is given in terms of the independent variables $E, N$ and $X_{i}$, all the thermodynamic variables can be generated by $S\left(E, N, X_{i}\right)$ using (2.9-2.11).

## 物理代写|统计物理代写Statistical Physics of Matter代考|Microcanonical Ensemble: Enumeration

The designation of microstates depends on the level of the description chosen. Let us consider a system composed $N$ interacting molecules. In the most microscopic level of the description, where the system is described quantum mechanically involving molecules and their subunits such as atoms and electrons, the microstates are the quantum states labeled by a simultaneously measurable set of quantum numbers of the system, which are virtually infinite. At the classical level of description, the microscopic states are specified by the $N$-particle phase space, i.e., the momenta and coordinates of all the molecules as well as their internal degrees of freedom. For both of these cases, enumeration of total number $W$ of microstates in a microcanonical ensemble would be a formidable task.

In many interesting situations, however, the description of the system need not be expressed in terms of the underlying quantum states or phase space. Consider a system that has $N$ distinguishable subunits, each of which can be in one of two states. A simple example is a linear array of $N$ sites each of which is either in the state 1 or 0 (Fig. 3.2a). Such two-state situations occur often in mesoscopic systems that lie between microscopic and macroscopic domains. The two state model not only allows the analytical calculation; although seemingly quite simple, it can be applied to many different, interesting problems of biological significance. Of particular interest are biological systems that consist of nanoscale subunits, for example (Fig. 3.2b) the specific sites in a biopolymer where proteins can bind via selective and non-covalent interactions and (c) the base-pairs in double-stranded DNA that can close or open.

Now, let us consider as our microcanonical system an array of $N$ such subunits (e.g., a biopolymer with $N$ binding sites, or a $N$-base DNA), each of which has two states with different energies. For simplicity we neglect the interaction between subunits. Due to thermal agitations the subunits undergo incessant transitions from an energy state to the other. What is the entropy of the array and what is the probability at which each state occurs in a subunit?

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Microstates and Entropy

$$P \mathcal{M}=\frac{1}{W\left(E, N, X_{i}\right)}$$

$$S\left(E, N, X_{i}\right)=k_{B} \ln W\left(E, N, X_{i}\right) .$$

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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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