## 物理代写|统计物理代写Statistical Physics of Matter代考|The Poiseuille Flow

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|The Poiseuille Flow

The flow of the fluid within a narrow cylindrical channel (tube) of radius $R$ (Fig. 19.5) is driven by a pressure gradient $\partial p / \partial z=-\Delta p / L$ along the $z$ axis. Equation (19.33) for the steady state in cylindrical $(r, z)$ coordinate is given by
$$\frac{\eta}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_z(r)}{\partial r}\right)=-\frac{\Delta p}{L} .$$

Multiplying the above by $r$ and integrating it over $r$, we have the equation
$$\frac{\partial u_z(r)}{\partial r}=-\frac{\Delta p}{2 \eta L}\left(r+\frac{c}{r}\right)$$
where the constant $c$ vanishs to assure a finite value for $\partial u_z(r) / \partial r$ at $r=0$. We note that the equation for the shear stress is
$$\sigma_{z r}=-\eta \partial u_z(r) / \partial r=\Delta p r /(2 L)$$
Integrating (19.54) subject to the no-slip $\mathrm{BC}, u_z(r=R)=0$, leads to the parabolic velocity profile
$$u_z(r)=-\frac{\Delta p}{4 \eta L}\left(r^2-R^2\right)$$
Using this, one can obtain the volume flow per unit time (volumetric flow rate) per length along the flow:
$$Q=\int_0^R d r 2 \pi r u_z(r)=\frac{\pi \Delta p}{8 \eta L} R^4,$$
This is the famous formula called the Hagen-Poiseuille’s law.

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Low Reynolds Number Approximation

In the Navier-Stokes equation, there are two competing terms, the nonlinear inertia term $\rho \boldsymbol{u} \cdot \nabla \boldsymbol{u}$ and the viscous dissipation term $\eta \nabla^2 \boldsymbol{u}$. The ratio of the inertia term to the viscous term is called the Reynolds number: $\operatorname{Re}=|\rho \boldsymbol{u} \cdot \nabla \boldsymbol{u}| /\left|\eta \nabla^2 \boldsymbol{u}\right| \approx \rho U R / \eta$, where $U$ and $R$ are characteristic velocity and characteristic length of the flow. If $R e$ is above a certain critical value so that the nonlinear inertia term is important, the flow tends to be unpredictable, called turbulent. The turbulence is important in many practical problems such as large scale weather predictions and airplane designs, but its fundamental understanding has remained a long standing problem in physics.

If the $R e$ is lower than 1 so that the viscous term dominates over the nonlinear term, the flow tends to be laminar. The laminar flow is mathematically more tractable. Furthermore for the flows of biological organisms or complexes (of small $R$ ) in overdamping and viscous fluids (of high $\eta$ ), the laminar or low Reynolds number flows will be relevant. For example a bacterium of $1 \mu \mathrm{m}$ diameter that swims in water with a velocity of $2 \mu \mathrm{m}$ per second has the $\operatorname{Re} \approx 10^{-5}$. In this case the Navier-Stokes equation is simplified to equations for flow velocity
$$\nabla \cdot \boldsymbol{u}=0$$
and
$$\rho \frac{\partial}{\partial t} \boldsymbol{u}=-\nabla \cdot \boldsymbol{\sigma}=-\nabla p+\eta \nabla^2 \boldsymbol{u}$$
In the steady state the above becomes the Stokes equation
$$\nabla \cdot \boldsymbol{\sigma}=\nabla p-\eta \nabla^2 \boldsymbol{u}=0$$
which we study below.

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Poiseuille Flow

$$\frac{\eta}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_z(r)}{\partial r}\right)=-\frac{\Delta p}{L} .$$

$$\frac{\partial u_z(r)}{\partial r}=-\frac{\Delta p}{2 \eta L}\left(r+\frac{c}{r}\right)$$

$$\sigma_{z r}=-\eta \partial u_z(r) / \partial r=\Delta p r /(2 L)$$

$$u_z(r)=-\frac{\Delta p}{4 \eta L}\left(r^2-R^2\right)$$

$$Q=\int_0^R d r 2 \pi r u_z(r)=\frac{\pi \Delta p}{8 \eta L} R^4$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Low Reynolds Number Approximation

$$\nabla \cdot \boldsymbol{u}=0$$

$$\rho \frac{\partial}{\partial t} \boldsymbol{u}=-\nabla \cdot \boldsymbol{\sigma}=-\nabla p+\eta \nabla^2 \boldsymbol{u}$$

$$\nabla \cdot \boldsymbol{\sigma}=\nabla p-\eta \nabla^2 \boldsymbol{u}=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|统计物理代写Statistical Physics of Matter代考|Boltzmann Equation Explains Transport Equations

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|Boltzmann Equation Explains Transport Equations

Solutions to the transport equations above can be used to describe a variety of hydrodynamic phenomena such as diffusion, laminar flow, and heat transport. Fundamentally, these macroscopic transport equations are derived from a microscopic kinetic equations, e.g., the Boltzmann equation for the case of dilute gas. The Boltzmann equation is an equation for evolution of the probability density of a particle at velocity $\boldsymbol{v}$ and position $\boldsymbol{r}$, which reads
$$\frac{\partial f(\boldsymbol{r v}, t)}{\partial t}+\boldsymbol{v} \cdot \nabla f(\boldsymbol{r v}, t)=J(f f)$$
Here $J(f f)$ denotes the so-called collision integral that describes the temporal change of the probability density caused by two-particle collisions. The hydrodynamic densities, which are proportional to the velocity moments of $f(\boldsymbol{r} v, t)$, e.g., $\rho \boldsymbol{u}=m \int d \boldsymbol{v} \boldsymbol{v} f(\boldsymbol{r v}, t)$, are shown to satisfy the continuity equations. It took a long time before Chapmann and Enskog formulated the Boltzmann equation’s particular solutions to derive the hydrodynamic transport equations along with the transport coefficients therein in terms of molecular parameters and collision mechanics of two interacting particles. The derivations of Boltzmann equation and the transport phenomena of gases therefrom mark an important page in history of non-equilibrium statistical mechanics.

One important feature of the transport phenomena is the time irreversibility. Consider that the particles initially confined in a volume freely diffuse to a region of lower particle density. The process is irreversible; in the lifetime of universe, the particle will never get back into the initial volume, by the second law of thermodynamics. The irreversibility can be seen from the diffusion equation for the density, $\partial n(r, t) / \partial t=D \nabla^2 n(r, t)$, which is not invariant with respect to the time reversal operation, $t \rightarrow-t$, but becomes $-\partial n(r,-t) / \partial t=D \nabla^2 n(r,-t)$. Because of the impossibility of this equation, the time reversed motion is not natural. There is only one direction, time arrow, from the past to the future. But look at the more fundamental, microscopic equation of the motion for the constituent particles, that is, the Newton’s equation, $m d v_i / d t=\boldsymbol{F}\left{r_j\right}$ for all particles labeled as $i$. This equation is invariant with respect to time reversal upon which $v_i \rightarrow-v_i$. Indeed a “time-backward” trajectory cannot be distinguished from a “time-forward” trajectory; the particles move just as well “backwards” as they do “forwards”. This is fundamentally at odds with the natural phenomena we observe macroscopically! The problem is called the time irreversibility paradox.

## 物理代写|统计物理代写Statistical Physics of Matter代考|A Simple Shear and Planar Flow

Consider a fluid between two large plates, each with an area $A$, separated by a distance $D$. The upper plate is in steady motion at a constant velocity $V$, while the other is at rest (Fig. 19.3). The fluid undergoes a shear flow (called the Couette flow) along $z$-axis on a $(y, z)$ plane, $\boldsymbol{u}=u_z(x) \hat{z}$, causing the stress $\sigma_{x z}=-\eta \partial u_z / \partial x$. The above relations reduce (19.33) to a remarkably simple form
$$\rho \frac{\partial}{\partial t} u_z(x, t)=-\frac{\partial p}{\partial z}+\eta \frac{\partial^2 u_z}{\partial x^2}$$
Furthermore, in the Couette flow situation, the pressure is uniform along $z$-direction, so
$$\rho \frac{\partial u_z}{\partial t}=\eta \frac{\partial^2 u_z}{\partial x^2}$$
The velocity $u_z$ satisfies the diffusion equation, akin to the equation which we already studied for the mass and heat diffusions.

P19.6 Consider an unbounded fluid above a plane at $x=0$ that moves in the $z$ direction with a time dependent velocity $V(t)=V_0 \cos \omega t$. Show that the fluid velocity for $x>0$ is given by
$$u_z(t)=V_0 \cos \left{\omega t-\left(\frac{\omega \rho}{2 \eta}\right)^{1 / 2} x\right} \exp \left{-\left(\frac{\omega \rho}{2 \eta}\right)^{1 / 2} x\right} .$$
In a steady state (19.45) is
$$\eta \frac{\partial^2 u_z}{\partial x^2}=0$$
This equation is to be solved subject to two $\mathrm{BC}$, usually the no slip $\mathrm{BC}$, according to which the fluid velocity on a surface is same as that of the surface: $u_z=V$ at $x=D$ and $u_z=0$ at $x=0$. Thus we find the solution
$$u_z=\frac{V}{D} x$$
which shows that the fluid velocity is sheared at a uniform rate $V / D$ along the $z$ direction.

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Boltzmann Equation Explains Transport Equations

$$\frac{\partial f(\boldsymbol{r} \boldsymbol{v}, t)}{\partial t}+\boldsymbol{v} \cdot \nabla f(\boldsymbol{r} \boldsymbol{v}, t)=J(f f)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|A Simple Shear and Planar Flow

$$\rho \frac{\partial}{\partial t} u_z(x, t)=-\frac{\partial p}{\partial z}+\eta \frac{\partial^2 u_z}{\partial x^2}$$

$$\rho \frac{\partial u_z}{\partial t}=\eta \frac{\partial^2 u_z}{\partial x^2}$$

P19.6 考虑平面上方的无界流体 $x=0$ 在 $z$ 方向与时间相关的速度 $V(t)=V_0 \cos \omega t$. 表明流体速度为 $x>0$ 是 (谁) 给的

$$\eta \frac{\partial^2 u_z}{\partial x^2}=0$$

$$u_z=\frac{V}{D} x$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## PHYS6562 Statistical Physics课程简介

The course presumes a high level of sophistication, equivalent to but not necessarily the same as that of a first-year physics graduate student (undergrad-level quantum, classical mechanics, and thermodynamics). Only a small portion of the course (roughly one and a half weeks) will demand a knowledge of quantum mechanics; students with no quantum background have found the rest of the course comprehensible and useful, if challenging. Primarily for graduate students.

## PREREQUISITES

The course presumes a high level of sophistication, equivalent to but not necessarily the same as that of a first-year physics graduate student (undergrad-level quantum, classical mechanics, and thermodynamics). Only a small portion of the course (roughly one and a half weeks) will demand a knowledge of quantum mechanics; students with no quantum background have found the rest of the course comprehensible and useful, if challenging. Primarily for graduate students.

J. Sethna.

A broad, graduate level view of statistical mechanics, with applications to not only physics and chemistry, but to computation, mathematics, dynamical and complex systems, and biology. Some traditional focus areas will not be covered in detail (thermodynamics, phase diagrams, perturbative methods, interacting gasses and liquids).

## PHYS6562 Statistical Physics HELP（EXAM HELP， ONLINE TUTOR）

A dynamical state of a biopolymer undergoes transition between two states 1 and 2 , with rates given by $R_{1 \rightarrow 2}=C e^{-\Delta f_{12} / k_6 T}$ and $R_{2 \rightarrow 1}=C e^{-\Delta f_{21} / k_B T}$, where $\Delta f_{i j}$ is the free energy barrier for the transitions. Suppose that $\Delta f_{12}=\Delta e-T \Delta s, \Delta f_{21}=$ De where $\Delta e>0$ and $\Delta s$ is the internal energy and entropy changes that are independent of temperature T. Apply a weak oscillating force of the frequency $\omega$ that couples with the reaction coordinate. If the force induces a stochastic resonance, what is the optimal value for the noise strength $k_B T$ ? How does it depend on the entropy changes $\Delta s$ ?

Show that the pressure of a stationary ideal gas satisfying $p_0=\rho k_B T / \mathrm{m}$ at $x$ under a uniform gravity $g$ along the $x$-axis is given by
$$p(x)=p_0 \exp \left(-m g x /\left(k_B T\right)\right)$$

A flexible polymer is anchored on planar surface at one end is subject to a Couette flow with a shear rate $\dot{\gamma}$, as shown in the figure below. Study the conformation of the chain in a steady state by finding (i) the mean square of EED $\left\langle\boldsymbol{R}^2\right\rangle$. where the average is taken over the steady state. (ii) Find $\langle\boldsymbol{r}(s, t)\rangle$.

Use the scaling argument above to show that the $C M$ diffusion constant and EED relaxation time for the chain in good solvents with $R_g \sim N^v$ are given by
$$D_c \sim N^{-v} \quad \text { and } \quad \tau_Z \sim N^{3 v}$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供cornell.edu PHYS6562 Statistical Physics统计物理的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|Macroscopic Consideration

In Chap. 9, we studied the static linear response theory, in which the change of a systems’ variable $\Delta X_i$ caused by a small static force or field $f_i$ conjugate to the variable is given by its fluctuation $\left\langle\left(\Delta \mathcal{X}_i\right)^2\right\rangle_0$. For example, the change in average extension of an elastic rod $\Delta X$ in response to a small applied tension $f$ is given by $\Delta X=\chi_s f$, where a constant $\chi_s$ is the static response function given by the fluctuation of the microscopic extension $\mathcal{X}$ at equilibrium in the absence of the force, $\chi_s=\beta\left\langle(\Delta \mathcal{X})^2\right\rangle_0$. The response function here is called stretch modulus.

Here we generalize the theory for the time-dependent situations questioning: how will the elastic rod extend dynamically in response to a small force acting on the system $f(t)$, which has an arbitrary time dependence? A naïve generalization may suggest $\Delta X(t)=\chi f(t)$, or $\Delta X(t)=\chi(t) f(t)$, either of which is wrong! Considering the linearity with respect to $f(t)$, we can deduce that the true relation is
$$\Delta X(t)=\int_{-\infty}^t \chi\left(t, t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime}$$
$\chi\left(t, t^{\prime}\right)$ is a time-dependent dynamic response function which is an intrinsic property of the system at the unperturbed state. Because the property is invariant with respect to time-translation, $\chi$ only depends on the difference $t-t^{\prime}$ connecting the response $\Delta X(t)$ and the cause $f\left(t^{\prime}\right): \chi\left(t, t^{\prime}\right)=\chi\left(t-t^{\prime}\right)$. Equation (17.1) signifies that system’s response to the force in general is delayed. Only in the limit $\chi\left(t-t^{\prime}\right) \rightarrow \chi_s \delta\left(t-t^{\prime}\right)$, the response is instantaneous, $\Delta X(t)=\chi_s f(t)$. Second,$\chi\left(t-t^{\prime}\right)$ is non-vanishing only when $t>t^{\prime}$, dictated by the principle of causality that the effect follows the cause. Thus (17.1) can be replaced by
$$\Delta X(t)=\int_{-\infty}^{\infty} \chi\left(t-t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime}$$
The linear response $\Delta X(t)$ to an oscillatory force $f\left(t^{\prime}\right)=a \cos \Omega t^{\prime}=\operatorname{Re}\left[a e^{-i \Omega t^{\prime}}\right]$ reads
\begin{aligned} \Delta X(t) & =\int_{-\infty}^t d t^{\prime} \chi\left(t-t^{\prime}\right) \operatorname{Re}\left[a e^{-i \Omega t^{\prime}}\right] \ & =\operatorname{Re}\left[\int_{-\infty}^t d t^{\prime} \chi\left(t-t^{\prime}\right) a e^{i \Omega\left(t-t^{\prime}\right)} e^{-i \Omega t}\right] \ & =\operatorname{Re}\left[\int_0^{\infty} d s \chi(s) a e^{i \Omega s} e^{-i \Omega t}\right]=\operatorname{Re}\left[\chi(\Omega) e^{-i \Omega t}\right], \end{aligned}
where
$$\chi(\Omega)=\int_0^{\infty} d t e^{i \Omega t} \chi(t)=\int_{-\infty}^{\infty} d t e^{i \Omega t} \chi(t),$$
is a time-Fourier transform of $\chi(t)$, which vanishes for $t<0$.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Statistical Mechanics of Dynamic Response Function

Now let us obtain $\chi(t)$ using statistical mechanics based on the microscopic view, for a stepwise unloading of $f_i$, which is not limited to the tension but can include a variety of forces and fields. Conjugate to $f_i$ is the system variable $\mathcal{X}_i$, whose average can not only be the macroscopic displacement $X_i$ introduced in (Table 2.1) but also be mesoscopic variables, e.g., the displacement of a Brownian particle.
We consider that from the distant past our system, viewed as a classical many-body system, is brought to an equilibrium state under a constant force $f_i$ until $t=0$, after which the force is turned off. At $t=0$ (initially), the system’s Hamiltonian is
$$\mathcal{H}(\Gamma(0))=\mathcal{H}0(\Gamma(0))-f_i \mathcal{X}_i(\Gamma(0))$$ where $\Gamma(0)$ is the systems’ many-particle phase space point descriptive of the initial state and evolves to $\Gamma(t)$ at a later time $t$ (Fig. 17.2b). The macroscopic displacement $X_j(t)$ at $t$ is the average of the corresponding microscopic variable of the system $\mathcal{X}_j(t)=\mathcal{X}_j(\Gamma(t))$ over all microstates initially prepared with the distribution $e^{-\beta \mathcal{H}(\Gamma(0))} / \sum{\mathcal{M}} e^{-\beta \mathcal{H}(\Gamma(0))}$
$$X_j(t)=\left\langle\mathcal{X}j(t)\right\rangle=\frac{\int d \Gamma(0)\left{\mathcal{X}_j(\Gamma(t)) e^{-\beta \mathcal{H}(\Gamma(0))}\right}}{\int d \Gamma(0) e^{-\beta \mathcal{H}(\Gamma(0))}}$$ Because $\mathcal{H}^{\prime}=-f_i \mathcal{X}_i$ is a perturbation, $e^{-\beta \mathcal{H}(\Gamma(0))} \approx e^{-\beta \mathcal{H}_0}\left(1+\beta f_i \mathcal{X}_i(0)\right)$, and \begin{aligned} \left\langle\mathcal{X}_j(t)\right\rangle & \approx \frac{\int d \Gamma(0)\left{\mathcal{X}_j(\Gamma(t)) e^{-\beta \mathcal{H}_0}\left(1+\beta f_i \mathcal{X}_i(0)\right)\right}}{\int d \Gamma(0) e^{-\beta \mathcal{H}_0}\left(1+\beta f_i \mathcal{X}_i(0)\right)} \ & =\frac{\left\langle\mathcal{X}_j(t)\right\rangle_0+\beta f_i\left\langle\mathcal{X}_j(t) \mathcal{X}_i(0)\right\rangle_0}{1+\beta f_i\left\langle\mathcal{X}_i(0)\right\rangle_0} \end{aligned} where $\langle\cdots\rangle_0$ is the average over the equilibrium ensemble in the absence of the force with the distribution $e^{-\beta \mathcal{H}_0} / \int d \Gamma(0) e^{-\beta \mathcal{H}_0(\Gamma(0))}$. Because, for time $t>0$, the perturbation is turned off and the time evolution is generated by $\mathcal{H}_0,\left\langle\mathcal{X}_j(t)\right\rangle_0 \equiv$ $\left\langle\mathcal{X}_j(\Gamma(t))\right\rangle_0$ is equal to $\left\langle\mathcal{X}_j(0)\right\rangle_0 \equiv\left\langle\mathcal{X}_j\right\rangle_0$, which is time-independent. If we retain in (17.17) the term linear in $f_i$, which is small, we arrive at an important result: \begin{aligned} \Delta X_j(t) & \equiv\left\langle\mathcal{X}_j(t)\right\rangle-\left\langle\mathcal{X}_j\right\rangle_0 \ & =\beta f_i\left\langle\Delta \mathcal{X}_j(t) \Delta \mathcal{X}_i(0)\right\rangle_0=\beta f_i C{j i}(t) \end{aligned}

# 统计物理代考

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

$\chi\left(t, t^{\prime}\right)$ 是时间相关的动态响应函数，它是系统在末受扰动状态下的固有属性。因为该属性对于时间平移 是不变的， $\chi$ 只取决于差异 $t-t^{\prime}$ 连接响应 $\Delta X(t)$ 和原因 $f\left(t^{\prime}\right): \chi\left(t, t^{\prime}\right)=\chi\left(t-t^{\prime}\right)$. 等式 (17.1) 表 示系统对一般力的响应是延迟的。只在极限 $\chi\left(t-t^{\prime}\right) \rightarrow \chi_s \delta\left(t-t^{\prime}\right)$ ，响应是瞬时的，
$\Delta X(t)=\chi_s f(t)$. 第二， $\chi\left(t-t^{\prime}\right)$ 仅当 $t>t^{\prime}$ ，遵循因果关系原则，即因果关系。因此 (17.1) 可以替 换为
$$\Delta X(t)=\int_{-\infty}^{\infty} \chi\left(t-t^{\prime}\right) f\left(t^{\prime}\right) d t^{\prime}$$

$$\Delta X(t)=\int_{-\infty}^t d t^{\prime} \chi\left(t-t^{\prime}\right) \operatorname{Re}\left[a e^{-i \Omega t^{\prime}}\right] \quad=\operatorname{Re}\left[\int_{-\infty}^t d t^{\prime} \chi\left(t-t^{\prime}\right) a e^{i \Omega\left(t-t^{\prime}\right)} e^{-i \Omega t}\right]=\operatorname{Re}$$

$$\chi(\Omega)=\int_0^{\infty} d t e^{i \Omega t} \chi(t)=\int_{-\infty}^{\infty} d t e^{i \Omega t} \chi(t)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Statistical Mechanics of Dynamic Response Function

$$\Delta X_j(t) \equiv\left\langle\mathcal{X}_j(t)\right\rangle-\left\langle\mathcal{X}_j\right\rangle_0 \quad=\beta f_i\left\langle\Delta \mathcal{X}_j(t) \Delta \mathcal{X}_i(0)\right\rangle_0=\beta f_i C j i(t)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|统计物理代写Statistical Physics of Matter代考|Flux-Over Population Method

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|Flux-Over Population Method

Finding the MPFT being often problematic in some cases, much easier and more direct way is to find crossing rate via the flux over population method. As shown by Reimann et al. (1999), the rate calculated this way is equal to the inverse of the Kramers time. In this method, we visualize a steady state where particles are constantly injected into the region at the reflecting boundary with a uniform current $J$ and are annihilated at the absorbing boundary. The rate of crossing the barrier is obtained by
$$R=\frac{J}{\wp_s}$$
where $\wp_s$ is the probability of the particle residing within the region:
$$\wp_s=\int_{\Omega} d q P(q) .$$
We revisit the simplest problem of one-dimensional free diffusion between a reflecting wall at $x=0$, and an absorbing wall at $x=L$. Although the real situation may be unsteady, to use the flux-over-population method, we imagine as if that particles are constantly injected at $x=0$ to induce a steady current $J$. The solution of $D \partial^2 P / \partial x^2=0$, is $P=a x+b$ yielding $J=-D \partial P / \partial x=-D a$. The solution subject to the absorbing $\mathrm{BC}$ at $x=L$ is $P(x)=-J(x-L) / D$. Because pre-transitional probability is $\wp_s=\int_0^L d x P(x)=\left(J L^2\right) / 2 D$, the rate is $J / \wp_s=$ $2 D / L^2$, which is the inverse of the MFPT, $\tau_0=L^2 / 2 D$.

For the case with a potential, we start with the equation for a constant flux, $(15.44)$
$$J=-\mathcal{D}(q) e^{-\Phi} \frac{\partial}{\partial q}\left(e^{\Phi} P\right)$$
which is integrated to:
$$J \int_{q_A}^q d q^{\prime} e^{\Phi\left(q^{\prime}\right)} / \mathcal{D}\left(q^{\prime}\right)=-\left[e^{\Phi(q)} P(q)-e^{\Phi\left(q_A\right)} P\left(q_A\right)\right]$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Kramers Problem for Polymer

The dynamics of polymer crossing barriers is a basic problem in soft matter; it is also important in various biological applications such as polymer transport across membranes and within channels, DNA gel electrophoresis, etc. We consider that each segment of the polymer is subject to a piece-wise harmonic potential $U(x)$ (Fig. 16.3) such that the distance between well bottom and barrier top is larger than the polymer’s radius of gyration. How can the Kramers rate (16.35) for a Brownian particle be extended to the linear chain of $N$ beads each with the same friction coefficient $\gamma$ ?

First suppose that a flexible polymer crosses the barrier in globular conformation. For the globule, we can adopt the single particle rate (16.35) with rescaling $U(x) \rightarrow N U(x)$ and thus $\omega_m \rightarrow N^{1 / 2} \omega_m, \omega_M \rightarrow N^{1 / 2} \omega_M$, as well as $\gamma \rightarrow N \gamma$ neglecting the hydrodynamic interactions between the beads, and find the crossing rate:

$$R_0=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta N \Delta U}$$
Compared with the single bead case, the prefactor $\left(\omega_m \omega_M\right) / 2 \pi \gamma$ remains unchanged whereas the activation energy is multiplied by $N$ times: the crossing rate of the polymer in globular state is vanishingly small.

Now consider that the polymer in crossing the barrier is unfolded into a flexible chain. With the reaction coordinate chosen to be the center of mass $(\mathrm{CM})$ of the chain, $X$, we then expect the rate to be modified to
$$R=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta \Delta \mathcal{F}}=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta\left(N \Delta U+\Delta \mathcal{F}^{\prime}\right)}$$
Here $\Delta \mathcal{F}=N \Delta U+\Delta \mathcal{F}^{\prime}$ is the free energy barrier for the chain to surmount, $\Delta \mathcal{F}^{\prime}=\mathcal{F}_M-\mathcal{F}_m$, where $\mathcal{F}_M, \mathcal{F}_m$ are the polymer conformational free energies with its CM fixed at the barrier top and well bottom, respectively. The free energy barrier $\Delta \mathcal{F}$ is much less than $N \Delta U$, due to the polymer flexibility, as will be shown below. Equation (16.37) was derived on the basis of multidimensional barrier crossing theory applied to $N$ beads interconnected by harmonic springs (Park and Sung 1999). The detailed derivation and expressions for $\mathcal{F}_M$ and $\mathcal{F}_m$ are quite involved, so here we present simple scaling theory arguments for long chains.
With the center of mass positioned at the well bottom, the flexible chain experiences confinement within the harmonic well, costing the conformational free energy, which is the sum of harmonic energy and the confinement-induced entropic contribution $(10.122)$ :
$$\mathcal{F}_m \sim \frac{1}{2} N \omega_m^2 \xi^2+\left(\frac{R_G}{\xi}\right)^2 k_B T$$

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Flux-Over Population Method

$$R=\frac{J}{\wp_s}$$

$$\wp_s=\int_{\Omega} d q P(q)$$

$$J=-\mathcal{D}(q) e^{-\Phi} \frac{\partial}{\partial q}\left(e^{\Phi} P\right)$$

$$J \int_{q_A}^q d q^{\prime} e^{\Phi\left(q^{\prime}\right)} / \mathcal{D}\left(q^{\prime}\right)=-\left[e^{\Phi(q)} P(q)-e^{\Phi\left(q_A\right)} P\left(q_A\right)\right]$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Kramers Problem for Polymer

$$R_0=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta N \Delta U}$$

$$R=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta \Delta \mathcal{F}}=\frac{\omega_m \omega_M}{2 \pi \gamma} e^{-\beta\left(N \Delta U+\Delta \mathcal{F}^{\prime}\right)}$$

$$\mathcal{F}_m \sim \frac{1}{2} N \omega_m^2 \xi^2+\left(\frac{R_G}{\xi}\right)^2 k_B T$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## PHYS6562 Statistical Physics课程简介

The course presumes a high level of sophistication, equivalent to but not necessarily the same as that of a first-year physics graduate student (undergrad-level quantum, classical mechanics, and thermodynamics). Only a small portion of the course (roughly one and a half weeks) will demand a knowledge of quantum mechanics; students with no quantum background have found the rest of the course comprehensible and useful, if challenging. Primarily for graduate students.

## PREREQUISITES

The course presumes a high level of sophistication, equivalent to but not necessarily the same as that of a first-year physics graduate student (undergrad-level quantum, classical mechanics, and thermodynamics). Only a small portion of the course (roughly one and a half weeks) will demand a knowledge of quantum mechanics; students with no quantum background have found the rest of the course comprehensible and useful, if challenging. Primarily for graduate students.

J. Sethna.

A broad, graduate level view of statistical mechanics, with applications to not only physics and chemistry, but to computation, mathematics, dynamical and complex systems, and biology. Some traditional focus areas will not be covered in detail (thermodynamics, phase diagrams, perturbative methods, interacting gasses and liquids).

## PHYS6562 Statistical Physics HELP（EXAM HELP， ONLINE TUTOR）

(1) The information entropy of a distribution $\left{p_n\right}$ is defined as $S=-\sum_n p_n \log _2 p_n$, where $n$ ranges over all possible configurations of a given physical system and $p_n$ is the probability of the state $|n\rangle$. If there are $\Omega$ possible states and each state is equally likely, then $S=\log _2 \Omega$, which is the usual dimensionless entropy in units of $\ln 2$.

Consider a normal deck of 52 distinct playing cards. A new deck always is prepared in the same order (A, $2 \boldsymbol{W} \cdots$ K $)$.
(a) What is the information entropy of the distribution of new decks?
(b) What is the information entropy of a distribution of completely randomized decks?
Now consider what it means to shuffle the cards. In an ideal riffle shuffle, the deck is split and divided into two equal halves of 26 cards each. One then chooses at random whether to take a card from either half, until one runs through all the cards and a new order is established (see figure).

(c) What is the increase in information entropy for a distribution of new decks that each have been shuffled once?
(d) Assuming each subsequent shuffle results in the same entropy increase (i.e. neglecting redundancies), how many shuffles are necessary in order to completely randomize a deck?
(e) If in parts (b), (c), and (d), you were to use Stirling’s approximation,
$$K ! \sim K^K e^{-K} \sqrt{2 \pi K}$$

(a) Since each new deck arrives in the same order, we have $p_1=1$ while $p_{2, \ldots, 52 !}=0$. Therefore $S=0$.
(b) For completely randomized decks, $p_n=1 / \Omega$ with $n \in{1, \ldots, \Omega}$ and $\Omega=52$ !, the total number of possible configurations. Thus, $S_{\text {random }}=\log 2 52 !=225.581$. (c) After one riffle shuffle, there are $\Omega=\left(\begin{array}{c}52 \ 26\end{array}\right)$ possible configurations. If all such configurations were equally likely, we would have $(\Delta S){\text {rifle }}=\log 2\left(\begin{array}{l}52 \ 26\end{array}\right)=48.817$. However, they are not all equally likely. For example, the probability that we drop the entire left-half deck and then the entire right half-deck is $2^{-26}$. After the last card from the left half-deck is dropped, we have no more choices to make. On the other hand, the probability for the sequence LRLR $\cdots$ is $2^{-51}$, because it is only after the $51^{\text {st }}$ card is dropped that we have no more choices. We can derive an exact expression for the entropy of the riffle shuffle in the following manner. Consider a deck of $N=2 K$ cards. The probability that we run out of choices after $K$ cards is the probability of the first $K$ cards dropped being all from one particular half-deck, which is $2 \cdot 2^{-K}$. Now let’s ask what is the probability that we run out of choices after $(K+1)$ cards are dropped. If all the remaining $(K-1)$ cards are from the right half-deck, this means that we must have one of the $\mathrm{R}$ cards among the first $K$ dropped. Note that this $\mathrm{R}$ card cannot be the $(K+1)^{\text {th }}$ card dropped, since then all of the first $K$ cards are $\mathrm{L}$, which we have already considered. Thus, there are $\left(\begin{array}{c}K \ 1\end{array}\right)=K$ such configurations, each with a probability $2^{-K-1}$. Next, suppose we run out of choices after $(K+2)$ cards are dropped. If the remaining $(K-2)$ cards are $\mathrm{R}$, this means we must have 2 of the $\mathrm{R}$ cards among the first $(K+1)$ dropped, which means $\left({ }^{K+1}{ }_2\right)$ possibilities. Note that the $(K+2)^{\text {th }}$ card must be $\mathrm{L}$, since if it were $\mathrm{R}$ this would mean that the last $(K-1)$ cards are $\mathrm{R}$, which we have already considered. Continuing in this manner, we conclude $$\Omega_K=2 \sum{n=0}^K\left(\begin{array}{c} K+n-1 \ n \end{array}\right)=\left(\begin{array}{c} 2 K \ K \end{array}\right)$$
and
$$S_K=-\sum_{a=1}^{\Omega_K} p_a \log 2 p_a=\sum{n=0}^{K-1}\left(\begin{array}{c} K+n-1 \ n \end{array}\right) \cdot 2^{-(K+n)} \cdot(K+n) .$$
The results are tabulated below in Table 1. For a deck of 52 cards, the actual entropy per riffle shuffle is $S_{26}=46.274$.
(d) Ignoring redundancies, we require $k=S_{\text {random }} /(\Delta S)_{\text {riftle }}=4.62$ shuffles if we assume all riffle outcomes are equally likely, and 4.88 if we use the exact result for the riffle entropy. Since there are no fractional shuffles, we round up to $k=5$ in both cases. In fact, computer experiments show that the answer is $k=9$. The reason we are so far off is that we have ignored redundancies, i.e. we have assumed that all the states produced by two consecutive riffle shuffles are distinct. They are not! For decks with asymptotically large

(2) In problem #1, we ran across Stirling’s approximation,
$$\ln K ! \sim K \ln K-K+\frac{1}{2} \ln (2 \pi K)+\mathcal{O}\left(K^{-1}\right),$$
for large $K$. In this exercise, you will derive this expansion.
(a) Start by writing
$$K !=\int_0^{\infty} d x x^K e^{-x},$$
and define $x \equiv K(t+1)$ so that $K !=K^{K+1} e^{-K} F(K)$, where
$$F(K)=\int_{-1}^{\infty} d t e^{K f(t)}$$
Find the function $f(t)$.
(b) Expand $f(t)=\sum_{n=0}^{\infty} f_n t^n$ in a Taylor series and find a general formula for the expansion coefficients $f_n$. In particular, show that $f_0=f_1=0$ and that $f_2=-\frac{1}{2}$.
(c) If one ignores all the terms but the lowest order (quadratic) in the expansion of $f(t)$, show that
$$\int_{-1}^{\infty} d t e^{-K t^2 / 2}=\sqrt{\frac{2 \pi}{K}}-R(K),$$
and show that the remainder $R(K)>0$ is bounded from above by a function which decreases faster than any polynomial in $1 / K$.
(d) For the brave only! – Find the $\mathcal{O}\left(K^{-1}\right)$ term in the expansion for $\ln K$ !.

(a) Setting $x=K(t+1)$, we have
$$K !=K^{K+1} e^{-K} \int_{-1}^{\infty} d t(t+1)^K e^{-t}$$
hence $f(t)=\ln (t+1)-t$
(b) The Taylor expansion of $f(t)$ is
$$f(t)=-\frac{1}{2} t^2+\frac{1}{3} t^3-\frac{1}{4} t^4+\ldots$$
(c) Retaining only the leading term in the Taylor expansion of $f(t)$, we have
\begin{aligned} F(K) & \simeq \int_{-1}^{\infty} d t e^{-K t^2 / 2} \ & =\sqrt{\frac{2 \pi}{K}}-\int_1^{\infty} d t e^{-K t^2 / 2} . \end{aligned}
Writing $t \equiv s+1$, the remainder is found to be
$$R(K)=e^{-K / 2} \int_0^{\infty} d s e^{-K s^2 / 2} e^{-K s}<\sqrt{\frac{\pi}{2 K}} e^{-K / 2},$$
which decreases exponentially with $K$, faster than any power.
(d) We have
\begin{aligned} F(K) & =\int_{-1}^{\infty} d t e^{-\frac{1}{2} K t^2} e^{\frac{1}{3} K t^3-\frac{1}{4} K t^4+\ldots} \ & =\int_{-1}^{\infty} d t e^{-\frac{1}{2} K t^2}\left{1+\frac{1}{3} K t^3-\frac{1}{4} K t^4+\frac{1}{18} K^2 t^6+\ldots\right} \ & =\sqrt{\frac{2 \pi}{K}} \cdot\left{1-\frac{3}{4} K^{-1}+\frac{5}{6} K^{-1}+\mathcal{O}\left(K^{-2}\right)\right} \end{aligned}
Thus,
$$\ln K !=K \ln K-K+\frac{1}{2} \ln K+\frac{1}{2} \ln (2 \pi)+\frac{1}{12} K^{-1}+\mathcal{O}\left(K^{-2}\right)$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供cornell.edu PHYS6562 Statistical Physics统计物理的代写代考辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## 物理代写|统计物理代写Statistical Physics of Matter代考|KYA322

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|A Brownian Motion Subject to a Harmonic Force

It is a challenge to analytically solve the Langevin equation with the external force in an arbitrary form. As a solvable important example, let us consider the one-dimensional case with $F=-k x$, and find $\Delta(t)=\left\langle x(t)^2\right\rangle$ as a function of time. The Langevin equation is written as
$$M x^{\prime \prime}+\zeta x^{\prime}+k x=f_R(t)$$
We multiply the above equation by $x$, and then average both sides. Noting that $\Delta^{\prime}=2\left\langle x x^{\prime}\right\rangle, \Delta^{\prime \prime}=2\left\langle x^{\prime 2}\right\rangle+2\left\langle x x^{\prime \prime}\right\rangle=\left(2 k_B T\right) / M+2\left\langle x x^{\prime \prime}\right\rangle$, we derive a differential equation for $\Delta(t)$ :
$$\frac{1}{2} \Delta^{\prime \prime}+\frac{1}{2} \tau_p^{-1} \Delta^{\prime}+\omega^2 \Delta=\frac{k_B T}{M},$$
where $\omega^2=k / M$.
P13.6 Derive (13.86). Why $\left\langle f_R(t) x(t)\right\rangle=0$ ?
In a long time, the system approaches to the equilibrium, and thus (13.86) reduces to $\Delta=k_B T /\left(M \omega^2\right)=k_B T / k$, which can also be derived from the equipartition of energy for the displacement $k\left\langle x^2\right\rangle / 2=k_B T / 2$. Defining $\delta=\Delta-k_B T / k,(13.86)$ becomes homogeneous,
$$\frac{1}{2} \delta^{\prime \prime}+\frac{1}{2} \tau_p^{-1} \delta^{\prime}+\omega^2 \delta=0$$
This is identical to the equation for a damped harmonic oscillator. Assuming the solution of the form $\delta \sim e^{-\lambda t}$, we find that, by substituting it in the equation above, there are two such $\lambda$ ‘s:

$$\lambda_{\pm}=\frac{1}{2 \tau_p}\left{1 \pm\left(1-8 \omega^2 \tau_p^2\right)^{1 / 2}\right}$$
The solution that satisfies the initial conditions, $\Delta=0, \Delta^{\prime}=0$ at $t=0$, is
$$\Delta(t)=\frac{k_B T}{k}\left(1-\frac{\lambda_{+} e^{-\lambda_{-} t}-\lambda_{-} e^{-\lambda_{+} t}}{\lambda_{+}-\lambda_{-}}\right)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Overdamped Langevin Equation

In many situations we deal with the behavior of a Brownian motion at times much longer than the velocity relaxation time $\tau_p$, where velocity or inertia of the particle becomes irrelevant. Excellent examples are colloids and macromolecules, where $\tau_p$ can be much smaller than the relevant time scale of the motions and conformational changes. In these cases the underdamped Langevin equation (13.64) is reduced to the overdamped Langevin equation
$$\zeta \frac{d x}{d t}=F(x)+f_R(t)$$
where $x$ may represent a position or certain conformational coordinate of interest, $f_R(t)$ is the Gaussian white noise given earlier. As will be shown later, this is the equation of motion equivalent to the Smoluchowski equation for the probability discussed earlier.
In the absence of an external force, (13.103) becomes
$$\zeta \frac{d x}{d t}=f_R(t)$$
This Langevin equation is equivalent to the diffusion equation. The stochastic dynamics of $x(t)$ is called the Wiener process. By integrating the equation above,$$x(t)=\frac{1}{\zeta} \int_0^t f_R\left(t^{\prime}\right) d t^{\prime}+x_0$$
one can confirm $\langle x(t)\rangle=x_0$ and
$$\left\langle\left(x(t)-x_0\right)^2\right\rangle=\frac{1}{\zeta^2} \int_0^t d t^{\prime} \int_0^t d t^{\prime \prime}\left\langle f_R\left(t^{\prime}\right) f_R\left(t^{\prime \prime}\right)\right\rangle=2 D t,$$
which is the Einstein displacement formula in one dimension.

## 物理代写|统计物理代写Statistical Physics of Matter代考|A Brownian Motion Subject to a Harmonic Force

$$M x^{\prime \prime}+\zeta x^{\prime}+k x=f_R(t)$$

$$\frac{1}{2} \Delta^{\prime \prime}+\frac{1}{2} \tau_p^{-1} \Delta^{\prime}+\omega^2 \Delta=\frac{k_B T}{M}$$

$\mathrm{P} 13.6$ 导出 (13.86)。为什么 $\left\langle f_R(t) x(t)\right\rangle=0 ?$

$$\frac{1}{2} \delta^{\prime \prime}+\frac{1}{2} \tau_p^{-1} \delta^{\prime}+\omega^2 \delta=0$$

$$\Delta(t)=\frac{k_B T}{k}\left(1-\frac{\lambda_{+} e^{-\lambda_{-} t}-\lambda_{-} e^{-\lambda_{+} t}}{\lambda_{+}-\lambda_{-}}\right)$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Overdamped Langevin Equation

$$\zeta \frac{d x}{d t}=F(x)+f_R(t)$$

$$\zeta \frac{d x}{d t}=f_R(t)$$

$$x(t)=\frac{1}{\zeta} \int_0^t f_R\left(t^{\prime}\right) d t^{\prime}+x_0$$

$$\left\langle\left(x(t)-x_0\right)^2\right\rangle=\frac{1}{\zeta^2} \int_0^t d t^{\prime} \int_0^t d t^{\prime \prime}\left\langle f_R\left(t^{\prime}\right) f_R\left(t^{\prime \prime}\right)\right\rangle=2 D t,$$

## 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。

## 物理代写|统计物理代写Statistical Physics of Matter代考|PHYSICS7546

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|A Trapped Brownian Particle

Hitherto in this section we were considering mostly the steady state diffusive motion of Brownian particles. Below we study the time-dependent motion of a Brownian particle in one dimension confined within a trap of length $L$. Whenever the particle arrives on the boundary $x=0$ or $L$, it is absorbed. If it is initially released at $x=x_0$, what is the probability density with which it is found at a position $x$ within the trap at a later time? What is the average time in which it will reside within the trap? You might imagine a drunken bug within a trap.
The diffusion equation for the probability density is written as
$$\frac{\partial P(x, t)}{\partial t}=-\mathcal{L} P(x, t)$$
where $\mathcal{L}=-D \partial^2 /\left(\partial x^2\right)$ is a linear operator. The solution is formally written as
$$P\left(x, t \mid x_0\right)=e^{-\mathcal{L} t} P\left(x_0, 0\right)=e^{-\mathcal{L} t} \delta\left(x-x_0\right)$$
Consider a set of eigenfunctions $\psi_n$ and eigenvalues $\lambda_n$ :
$$\mathcal{L} \psi_n=\lambda_n \psi_n$$
Using the completeness of the eigenfunctions, $\delta\left(x-x_0\right)=\sum_n \psi_n(x) \psi_n\left(x_0\right)$, (13.53) becomes
$$P\left(x, t \mid x_0\right)=\sum_n e^{-\lambda_n t} \psi_n(x) \psi_n\left(x_0\right) .$$
Subject to the boundary conditions at $x=0$ and $L$ where $\psi_n=0$, they are
$$\psi_n(x)=\left(\frac{2}{L}\right)^{1 / 2} \sin \frac{n \pi}{L} x, \quad \lambda_n=\left(\frac{n \pi}{L}\right)^2 D$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Velocity Langevin Equation

The Langevin equation is simply obtained by replacing the drift velocity $V$ of a Brownian particle in the macroscopic deterministic equation (13.13) by a fluctuating velocity $v$, and adding to the right hand side a fluctuation term $f_R(t)$ called the random force. Considering $1-D$ motion for simplicity, the Langevin equation is written as:
$$M \frac{d v}{d t}=-\zeta v+F(x)+f_R(t)$$

The fluctuating force $f_R(t)$ is due to the collisions of surrounding fluid molecules with the Brownian particle that are not incorporated in the frictional force $-\zeta v$. Since the Brownian particle is much heavier than a fluid molecule, the random force $f_R(t)$ is supposed to vary irregularly and rapidly on the timescale of the velocity. $f_R(t)$ can be constructed as a sum of many contributions from surrounding fluid molecules at different times, each of which is not correlated with other on the timescale. Then the Central Limit Theorem (Chap. 10) tells us that the random force is distributed in Gaussian prescribed solely by the first two moments. The first one is the average, which, due to the randomness, vanishes:
$$\left\langle f_R(t)\right\rangle=0$$
The second moment is expressed as
$$\left\langle f_R(t) f_R\left(t^{\prime}\right)\right\rangle=2 \Theta \delta\left(t-t^{\prime}\right)$$
The averages are taken over the equilibrium ensemble. On the time scale of the velocity, random force fluctuates very rapidly and does not correlate with itself at different times. This delta-function-correlated random force is called the white noise, because the Fourier transform of (13.66), which is called the power spectrum of the random force, is independent of the frequency. This Gaussian and white noise is called thermal noise; the constant $\Theta$ is the strength of the noise, which will be shown to be $\zeta k_B T$ shortly. The Langevin equation (13.64) with this non-analytic noise term is an example of the stochastic differential equation.

## 物理代写|统计物理代写Statistical Physics of Matter代考|A Trapped Brownian Particle

$$\frac{\partial P(x, t)}{\partial t}=-\mathcal{L} P(x, t)$$

$$P\left(x, t \mid x_0\right)=e^{-\mathcal{L} t} P\left(x_0, 0\right)=e^{-\mathcal{L} t} \delta\left(x-x_0\right)$$

$$\mathcal{L} \psi_n=\lambda_n \psi_n$$

$$P\left(x, t \mid x_0\right)=\sum_n e^{-\lambda_n t} \psi_n(x) \psi_n\left(x_0\right) .$$

$$\psi_n(x)=\left(\frac{2}{L}\right)^{1 / 2} \sin \frac{n \pi}{L} x, \quad \lambda_n=\left(\frac{n \pi}{L}\right)^2 D$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Velocity Langevin Equation

Langevin 方程简单地通过替换漂移速度得到 $V$ 宏观确定性方程 (13.13) 中布朗粒子的波动速度 $v$, 并在右侧添加一个波动项 $f_R(t)$ 称为随机力。考虑 $1-D$ 为了简单起见，朗之万方程写为:
$$M \frac{d v}{d t}=-\zeta v+F(x)+f_R(t)$$

$$\left\langle f_R(t)\right\rangle=0$$

$$\left\langle f_R(t) f_R\left(t^{\prime}\right)\right\rangle=2 \Theta \delta\left(t-t^{\prime}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|统计物理代写Statistical Physics of Matter代考|PHYS7635

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|Polymer Binding–Unbinding

A polymer chain can bind to an attracting surface but, because of the free energy cost that the confinement incurs, it can also unbind from the surface. To study the polymer binding unbinding transition quantitatively, consider the surface is $(y, z)$ plane and the interaction between a polymer bead and surface given by the hard-square well potential, which is a simplest model characterized by potential depth $U_0$ and range $a$ as depicted in the Fig. 10.11:
$$u(x)= \begin{cases}\infty, & x=0 \ U_0, & 0a\end{cases}$$
where $x$ is the coordinate of the chain end vertical to the surface. Neglecting the lateral sonerdinates $y$ and $z_1$ along which the shain snd distrihution is Caussian, it suffice to consider the one-dimensional Edwards equation,
$$-\frac{\partial}{\partial N} G\left(x, x^{\prime} ; N\right)=\left[-\frac{l^2}{6} \frac{\partial^2}{\partial x^2}+\beta u(x)\right] G\left(x, x^{\prime} ; N\right) .$$
The solution and its ground state dominance approximation is given as
\begin{aligned} G\left(x, x^{\prime} ; N\right) &=\sum_{n=0} e^{-N \epsilon_n} \psi_n(x) \psi_n\left(x^{\prime}\right) \ & \approx e^{-N \epsilon_0} \psi_0(x) \psi_0\left(x^{\prime}\right) . \end{aligned}

The ground state eigenfunction $\psi_0(x)$ and eigenvalue $\epsilon_0$ satisfy
$$\left[-\frac{l^2}{6} \frac{\partial^2}{\partial x^2}+\beta u(x)\right] \psi_0(x)=\epsilon_0 \psi_0(x)$$
$\psi_0(x)$ that satisfies the $\mathrm{BC}\left(\psi_0(x=0)=0, \psi_0(x \rightarrow \infty)=0\right)$ are given by
$$\psi_0(x)= \begin{cases}A \sin k x & x \geq a \ B e^{-k x} & x<a\end{cases}$$
where
$$k=\left{\frac{6}{l^2}\left(\beta U_0-\left|\epsilon_0\right|\right)\right}^{1 / 2}$$
and
$$\kappa=\left{\frac{6}{l^2}\left|\epsilon_0\right|\right}^{1 / 2}$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Polymer Exclusion and Condensation

The ideal chain model assumes that polymer segments can overlap, but due to the space they occupy, the real chain cannot cross itself, and thus cannot be modelled by a random walk but by a “self-avoiding walk”. This excluded volume effect allows the polymer coil to swell. But if this repulsive interaction is dominated by the attractive interaction between the segments, the coiled polymer undergoes a collapse transition into a condensed state called a polymer globule. Here we characterize the EED for various conformational states and study the conditions of the transitions between them.

As a measure of the overall conformation of the polymer, which is modulated by solvent, we study how the equilibrium end-to-end length $R$ depends on $N$. To this end we seek a chain’s free energy function of $R$ with $N$ fixed. First consider an ideal chain, where there are no inter-bead interactions other than incorporated in the chain connectivity. The probability distribution function (PDF) $D(R ; N)$ that the ideal chain’s end is within $d R$ is the EED PDF $P(\boldsymbol{R} ; N)$ times the volume element $d V$ taken to be spherical shell of radius $R$ and thickness $d R$ :
\begin{aligned} D(R ; N) d R &=P(\boldsymbol{R} ; N) d V \ &=\left(\frac{3}{2 \pi N l}\right)^{3 / 2} \exp \left(-\frac{3 R^2}{2 N l}\right) 4 \pi R^2 d R . \end{aligned}
The free energy $\mathcal{F}_0(R)$ of the ideal chain associated with $R$ is then given by,
\begin{aligned} \mathcal{F}_0(R) &=-k_B T \ln D(R ; N) \ &=k_B T\left(\frac{3}{2 N l^2} R^2-2 \ln R\right), \end{aligned}
apart from the part independent of $R$. Note that $\mathcal{F}_0(R)$ is different from $\mathcal{F}(\boldsymbol{R})$, (10.18), because here we are dealing with the degree of freedom, $Q=R$, not with $Q=\boldsymbol{R}$. The most probable (free-energy minimizing) value of $R$ is given by
$$R_p=\left(\frac{2}{3}\right)^{1 / 2} R_0 \sim N^{1 / 2},$$
which is on par with $R_0=N^{1 / 2} l$ as well as the free chain radius of gyration $R_G=(1 / 6)^{1 / 2} R_0$

## 物理代写|统计物理代写物质统计物理学代考|聚合物绑定-解绑定

.

$$u(x)= \begin{cases}\infty, & x=0 \ U_0, & 0a\end{cases}$$
，其中$x$是链端垂直于表面的坐标。忽略shain snd分布为高斯分布的横向声纳坐标$y$和$z_1$，就可以考虑一维Edwards方程
$$-\frac{\partial}{\partial N} G\left(x, x^{\prime} ; N\right)=\left[-\frac{l^2}{6} \frac{\partial^2}{\partial x^2}+\beta u(x)\right] G\left(x, x^{\prime} ; N\right) .$$
，其解及其基态优势近似为
\begin{aligned} G\left(x, x^{\prime} ; N\right) &=\sum_{n=0} e^{-N \epsilon_n} \psi_n(x) \psi_n\left(x^{\prime}\right) \ & \approx e^{-N \epsilon_0} \psi_0(x) \psi_0\left(x^{\prime}\right) . \end{aligned}

$$\left[-\frac{l^2}{6} \frac{\partial^2}{\partial x^2}+\beta u(x)\right] \psi_0(x)=\epsilon_0 \psi_0(x)$$
$\psi_0(x)$满足$\mathrm{BC}\left(\psi_0(x=0)=0, \psi_0(x \rightarrow \infty)=0\right)$由
$$\psi_0(x)= \begin{cases}A \sin k x & x \geq a \ B e^{-k x} & x<a\end{cases}$$

$$k=\left{\frac{6}{l^2}\left(\beta U_0-\left|\epsilon_0\right|\right)\right}^{1 / 2}$$

$$\kappa=\left{\frac{6}{l^2}\left|\epsilon_0\right|\right}^{1 / 2}$$

## 物理代写|统计物理代写物质统计物理学代考|聚合物排斥和缩合

. .

\begin{aligned} D(R ; N) d R &=P(\boldsymbol{R} ; N) d V \ &=\left(\frac{3}{2 \pi N l}\right)^{3 / 2} \exp \left(-\frac{3 R^2}{2 N l}\right) 4 \pi R^2 d R . \end{aligned}

\begin{aligned} \mathcal{F}_0(R) &=-k_B T \ln D(R ; N) \ &=k_B T\left(\frac{3}{2 N l^2} R^2-2 \ln R\right), \end{aligned}除了独立的部分 $R$。注意 $\mathcal{F}_0(R)$ 不同于 $\mathcal{F}(\boldsymbol{R})$，(10.18)，因为这里我们处理的是自由度， $Q=R$，不与 $Q=\boldsymbol{R}$。的最可能值(自由能最小值) $R$
$$R_p=\left(\frac{2}{3}\right)^{1 / 2} R_0 \sim N^{1 / 2},$$
，这与 $R_0=N^{1 / 2} l$ 以及旋转的自由链半径 $R_G=(1 / 6)^{1 / 2} R_0$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|统计物理代写Statistical Physics of Matter代考|KYA322

statistics-lab™ 为您的留学生涯保驾护航 在代写统计物理Statistical Physics of Matter方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计物理Statistical Physics of Matter代写方面经验极为丰富，各种代写统计物理Statistical Physics of Matter相关的作业也就用不着说。

• 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|The Chain Free Energy and Segmental Distribution

Once we find the polymer Green’s function, we can obtain the free energy function $\mathcal{F}\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)$ with its initial and final positions as the relevant degrees of freedom $Q=$ $\left(\boldsymbol{r}, \boldsymbol{r}^{\prime}\right)$ via the relation
$$e^{-\beta \mathcal{F}\left(\boldsymbol{r} r^{\prime}\right)} \propto G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)$$
The integration of (10.50) over $\boldsymbol{r}, \boldsymbol{r}^{\prime}$ yields the partition function of the chain,
$$Z_N \propto \int d \boldsymbol{r} \int d \boldsymbol{r}^{\prime} G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)$$
from which thermodynamic free energy $F(N)=-k_B T \ln Z_N$ is obtained. The proportionality in (10.57) will often be replaced by equality, without incurring any distinction in conformational and thermodynamic properties.

Because the $G\left(r, r^{\prime} ; N\right)$ is the probability density of the chain end located at the position $r$ given the initial point at $r^{\prime}$, the probability density of the end to be at $r$ regardless the location of the initial point is given by
$$\wp(\boldsymbol{r})=\int d \boldsymbol{r}^{\prime} G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right) / \int d \boldsymbol{r} \int d \boldsymbol{r}^{\prime} G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)$$
Now we make an approximation that is useful for a long chain, using the eigen-functions of the Edwards equation. For the case that the potential allows discrete bound states, the eigen-function expansion (10.46) for a long chain (large $N$ ) is dominated by the ground state labeled as $n=0$,
$$G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right) \approx e^{-N \epsilon_0} \psi_0(\boldsymbol{r}) \psi_0\left(\boldsymbol{r}^{\prime}\right)$$
This feature is owing to the reality of all the variables involved in the expansion, which is not possible for the corresponding Schrödinger equation.

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Effect of Confinemening a Flexible Chain

Suppose a free chain is brought within a box (Fig 10.8). Below we study the free energy of the confinement and the pressure of the chain on the walls following Doi and Edwards (1986).

The presence of the impenetrable wall is expressed by an infinite potential, $u(\boldsymbol{r})=\infty$, which can be implemented by the boundary condition $G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)=0$ for $\boldsymbol{r}$ and $\boldsymbol{r}^{\prime}$ on the wall, for the diffusion equation within the box:
$$\frac{\partial}{\partial N} G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)=\frac{l^2}{6} \nabla^2 G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right) .$$
First note that the Green’s function is separable into the Cartesian components,
$$G\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} ; N\right)=g_x\left(x, x^{\prime} ; N\right) g_y\left(y, y^{\prime} ; N\right) g_z\left(z, z^{\prime} ; N\right) .$$

Each component, for example, the $x$ component satisfies
$$\frac{\partial}{\partial N} g_x\left(x, x^{\prime} ; N\right)=\frac{l^2}{6} \frac{\partial^2}{\partial x^2} g_x\left(x, x^{\prime} ; N\right)$$
for which the Green’s function solution is
$$g_x\left(x, x^{\prime} ; N\right)=\sum_{n_x=1}^{\infty} e^{-N \epsilon_x} \psi_{n_x}(x) \psi_{n_x}\left(x^{\prime}\right)$$
The eigenfunctions and eigenvalues are
$$\psi_{n_x}(x)=\left(\frac{2}{L_x}\right)^{1 / 2} \sin \frac{n_x \pi x}{L_x}$$
and
$$\epsilon_{n_x}=\frac{l^2 n_x^2 \pi^2}{6 L_x}$$
respectively, where $n_x$ is the positive integers $1,2,3, \ldots$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。