Posted on Categories:Thermodynamics, 热力学, 物理代写

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## 物理代写|热力学代写Thermodynamics代考|Selective Decay

This rather frustrating series of failures may suggest that the problem lies in the common assumption of all the proposals discussed so far: LTE. For example, the equations of motion in many physical systems show that different quantities evolve with different time-scales and spatial scales; this result does not rely on LTE (explicitly, at least). Accordingly, it is at least conceivable that relaxation satisfies the ‘selective decay’ scenario.

The selective decay hypothesis is characterized by the following. If one considers the ‘ideal invariants’ of the system (namely, the quantities which would remain exactly constant during the evolution of the system should no dissipation occur), once dissipation has been introduced these quantities do not remain constant but start decaying, unless the external world somehow maintains their value constant. It is often found that one of these quantities is somehow ‘better conserved’ or ‘more rugged’ than others, i.e. that its typical decay time is much longer than the decay time of other quantities. ${ }^{11}$ If one minimizes the expression for the poorly conserved invariant subject to the constraint that the rugged invariant is conserved using the technique of Lagrange multipliers (Sect. A.3), an Euler-Lagrange equation for the field variables in the relaxed state results. The Lagrange multiplier is the ratio of the poorly conserved invariant to the ruggedly conserved one.

Typically, selective decay applies to problems in two-dimensional and threedimensional magnetohydrodynamics (‘MHD’), where the couples ‘rugged invariant versus poorly conserved invariant’ are ‘energy versus mean square vector potential’ and ‘energy versus magnetic helicity’, respectively. In MHD, for example, a well-known example of relaxed state is described by Taylor’s principle of minimum magnetic energy $\propto \int|\mathbf{B}|^2 d \mathbf{x}$ with fixed magnetic helicity $\int(\mathbf{A} \cdot \mathbf{B}) d \mathbf{x}$ (where $\mathbf{B}=\nabla \wedge \mathbf{A}$ and $\mathbf{A}$ is the vector potential) [25]. In Hall MHD, i.e. a macroscopic description of magnetized plasmas (made of two species, electrons and ions with ion mass $m_{i o n}$ and ion electric charge $q_{i o n}$ ) where electrons are effectively decoupled from ions, the couple ‘rugged invariant versus poorly conserved invariant’ is ‘total (magnetic + kinetic) energy’ versus ‘magnetic helicity and generalized ion helicity $\int(\mathbf{V} \cdot \boldsymbol{\Omega}) d \mathbf{x}$ ‘, with $\mathbf{V} \equiv \mathbf{v}+\frac{q_{\text {ion }}}{m_{\text {ion }}} \mathbf{A}$ and $\boldsymbol{\Omega} \equiv \nabla \wedge \mathbf{V}$. Taylor’s principle is replaced by Turner’s principle [26] of minimization of total energy with two constraints: fixed magnetic helicity and fixed generalized helicity. Remarkably, and in qualitative agreement with Kirchhoff’s principle of Sect. 5.3.1, in order to describe plasmas in the solar corona it has been postulated [27] to replace Turner’s principle with the constrained minimization of Joule heating power; fixed generalized ion helicity and its electron counterpart are the constraints. ${ }^{12}$

## 物理代写|热力学代写Thermodynamics代考|Maximal Entropy

The principle of ‘maximal entropy’13 dictates that the air in a room initially distributed in clumps moves towards smooth uniformity; thermodynamic equilibrium does not admit large-scale structures. However, for a system with a constrained phase space, maximal entropy can generate large-scale structures as a long-lived intermediate state. Remarkably, no LTE is explicitly invoked. To apply the principle of maximal entropy, one needs to consider a discrete or quantized version of the field variables. If we have $N$ such quanta of the field, ${ }^{14}$ we consider the number of ways these $N$ quanta can be arranged in a given state (like spins up or down). The most probable state is the one with the most permutations or the highest entropy subject to other constraints (such as conservation of energy and particle number); here entropy is defined as the logarithm of the number of permutations times Boltzmann constant. The description of the system is perfectly analogous to the familiar description of the $2 \mathrm{D}$ spin system in statistical mechanics of thermodynamic equilibrium [32]. The maximal entropy perspective addresses the question: are these observed large-scale, self-organized structures in some sense statistically more probable than other less simple ones?

Again, our room is a thermodynamically open system. The room exchanges either heat (across the closed window) or both heat and mass (across the open window) with the external world. In a relaxed state, the long-lived, large-scale structures are supposed to live not just for a long time, but indefinitely. The relevance of these approaches to the relaxed state of our room is, therefore, yet to be proven.

## 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:Thermodynamics, 热力学, 物理代写

## 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|热力学代写Thermodynamics代考|Entropy Production Due to Diffusion

We describe the diffusion ${ }^{49}$ from region 1 to region 2 located at different lengths in a region of cross section $\Sigma$ as a ‘reaction’ $1 \rightarrow 2$ where one particle disappears from region 1 (with length $l$ and volume $l \cdot \Sigma$ ) and appears in region 2-see Fig. 4.8.
Our formalism gives then $-v_1=\nu_2=1,-d n_1=d n_2=d \xi^{\prime}, T_1=T_2=T$ and $A=-v_1 \mu_1-v_2 \mu_2=\mu_1-\mu_2$. If diffusion is one-dimensional and occurs only in the $x$ direction perpendicular to the cross section with $j_{p x} x$-th component of the particle flow $\mathbf{j}p$, then volume integration on region 1 of the the balance of mass leads to: $\frac{d n_1}{d t} \cdot l \cdot \Sigma+j{p x} \cdot \Sigma=0$, hence $\frac{d n_1}{d t}=-\frac{j_{p x}}{l}$. It follows that $T \sigma=\Upsilon^{\prime} A=\frac{d \xi^{\prime}}{d t} A=-\frac{d n_1}{d t} A=\frac{j_{p x} A}{l}=-\frac{j_{p x}\left(\mu_2-\mu_1\right)}{l}$. Straightforward generalization to three dimensions for arbitarily small $l$ leads to $\sigma=-\mathbf{j}p \cdot \nabla\left(\frac{\mu}{T}\right)$. Moreover, if many species are present and the particle flow of the $k$-th species is $\mathbf{j}{p k} \equiv \frac{\mathbf{j}k}{m_k}$, further generalization gives: $$\sigma=-\mathbf{j}{p k} \cdot \nabla\left(\frac{\mu_k}{T}\right)=-\mathbf{j}_k \cdot \nabla\left(\frac{\mu_k^0}{T}\right)$$

What about LNET? The quantity $\nabla\left(\frac{\mu_k}{T}\right)$ is a gradient, and the divergence of $\mathbf{j}_{p k}$ is proportional to a time derivative. Moreover, Fick’s law ${ }^{50}$ is a linear relationship between these quantities. Finally, $\tau$ is related to the diffusion time-scale. We conclude that LNET holds. It is therefore far from surprising that Prigogine has proposed a MinEP for this problem [19]-see Sect. 4.3.10.

## 物理代写|热力学代写Thermodynamics代考|Fick’s Law

The thermodynamic fluxes and forces in the diffusion-related entropy production density $\sigma=-\mathbf{j}{p k} \cdot \nabla\left(\frac{\mu_k}{T}\right)$ are $\mathbf{j}{p k}$ and $-\nabla\left(\frac{\mu_k}{T}\right)$ respectively. Accordingly, if $\nabla T=$ 0 then the linear phenomenological laws ${ }^{52}$ are of the type $\mathbf{j}{p k}=-\frac{L{p k}}{T} \nabla \mu_k$. In a linear treatment of a system with $N$ species it is usually assumed that:
$$\mathbf{j}{p k}=-D{i k} \nabla n_k \quad ; \quad i, k=1 \ldots N$$

(‘Fick’s law’). If $N=1$ then $\nabla \mu=\frac{\partial \mu}{\partial n} \nabla n$ and Fick’s law implies $\mathbf{j}p=-D \nabla n$, hence $L{11}=\frac{T D}{\frac{\partial \mu}{\partial n}}$. If $N=2$ then it is customary to deal with the case $\nabla T=0$, $\nabla p=0$ for simplicity. As usual in such cases, we start with the condition of extremum of Gibbs’ free energy $G ; d G=0$ implies $\mu_k d n_k=0$. But $G=\mu_k n_k$, then $\mu_k d n_k=0$ implies $n_k d \mu_k=0$. Moreover, LTE at all times implies $d a=$ $\frac{d a}{d t} \cdot d t=\left(\frac{\partial a}{\partial t}+\mathbf{v} \cdot \nabla a\right) \cdot d t=\left(\frac{\partial a}{\partial t}+\frac{d \mathbf{r}}{d t} \cdot \nabla a\right) \cdot d t$ for the generic quantity $a$. In steady state $\frac{\partial}{\partial t}=0$, then $d a=d \mathbf{r} \cdot \nabla a$. Let $a=\mu_k$. Accordingly, $n_k d \mu_k=0$ implies $\left(n_k \nabla \mu_k\right) \cdot d \mathbf{r}=0$ for arbitrary $d \mathbf{r}$, hence:
$$0=n_k \nabla \mu_k=n_1 \nabla \mu_1+n_2 \nabla \mu_2$$
Correspondingly:
$$\sigma=-\mathbf{j}{p k} \cdot \nabla\left(\frac{\mu_k}{T}\right)=-\frac{\mathbf{j}{p k}}{T} \cdot \nabla \mu_k=-\frac{\mathbf{j}{p 1}}{T} \cdot \nabla \mu_1-\frac{\mathbf{j}{p 2}}{T} \cdot \nabla \mu_2=-\frac{1}{T}\left(\mathbf{j}{p 1}-\frac{n_1}{n_2} \mathbf{j}{p 2}\right) \cdot \nabla \mu_1$$
Further simplification follows from the fact that $d\left(\frac{1}{\rho}\right)=v_k d c_k$ for $\nabla T=0$ and $\nabla p=0$, as the definition of the mass flow $\mathbf{j}k$ for the $k$-th species implies $\rho \frac{d}{d t}\left(\frac{1}{\rho}\right)=$ $v_k \rho \frac{d c_k}{d t}=v_k \nabla \cdot \mathbf{j}_k$. Usually, diffusion leaves $\rho$ unaffected, as it is a slow, incompressible process. Then $v_k \nabla \cdot \mathbf{j}_k=0$. Moreover, $v_k \nabla \cdot \mathbf{j}_k=\nabla \cdot\left(v_k \mathbf{j}_k\right)-\mathbf{j}_k \cdot \nabla v_k=\nabla$. $\left(v_k \mathbf{j}_k\right)-\mathbf{j}_k \cdot \nabla\left(\frac{\partial \mu_k^0}{\partial p}\right)_T$ and $\mathbf{j}_k \cdot \nabla\left(\frac{\partial \mu_k^0}{\partial p}\right)_T=\mathbf{j}{p k} \cdot \nabla\left(\frac{\partial \mu_k}{\partial p}\right)T=\left[\frac{\partial\left(\mathbf{j}{p k} \cdot \nabla \mu_k\right)}{\partial p}\right]T \propto$ $\left[\frac{\partial\left(\mathbf{j}{p k} \cdot \nabla \mu_k\right)}{\partial p}\right]T=O\left(n_k \nabla \mu_k\right)=0$ because of $n_k \nabla \mu_k=0$. Consequently, incompressibility implies $\nabla \cdot\left(v_k \mathbf{j}_k\right)=0$. Together with the definitions of $\mathbf{j}{p k} \equiv \frac{\mathbf{j}k}{m_k}$ and of $v_k^{\prime} \equiv m_k v_k$, this gives ${ }^{53} \mathbf{j}{p 1} v_1^{\prime}+\mathbf{j}{p 2} v_2^{\prime}=0$. Then $$\sigma=-\frac{1}{T}\left(1+\frac{n_1 v_1^{\prime}}{n_2 v_2^{\prime}}\right) \mathbf{j}{p 1} \cdot \nabla \mu_1=-\frac{1}{T}\left(1+\frac{n_1 v_1^{\prime}}{n_2 v_2^{\prime}}\right)\left(\frac{\partial \mu_1}{\partial n_1}\right){p, T} \mathbf{j}{p 1} \cdot \nabla n_1$$

## 物理代写|热力学代写Thermodynamics代考|Entropy Production Due to Diffusion

$\frac{d n_1}{d t} \cdot l \cdot \Sigma+j p x \cdot \Sigma=0$, 因此 $\frac{d n_1}{d t}=-\frac{j_{\mu x}}{l}$. 它邅循 $T \sigma=\Upsilon^{\prime} A=\frac{d \xi^{\prime}}{d t} A=-\frac{d n_1}{d t} A=\frac{j_{\mu x} A}{l}=-\frac{j_{p x}\left(\mu_2-\mu_1\right)}{l}$. 任意小到三 个维度的简单概括 $l$ 导致 $\sigma=-\mathbf{j} p \cdot \nabla\left(\frac{\mu}{T}\right)$. 此外，如果存在许多物种并且粒子流 $k$-th 种是 $\mathbf{j} p k \equiv \frac{\mathrm{j} k}{m_k}$ ，进一步概括给出:
$$\sigma=-\mathbf{j} p k \cdot \nabla\left(\frac{\mu_k}{T}\right)=-\mathbf{j}k \cdot \nabla\left(\frac{\mu_k^0}{T}\right)$$ LNET呢? 数量 $\nabla\left(\frac{\mu_k}{T}\right)$ 是一个梯度，而散度 $\mathbf{j}{p k}$ 与时间导数成正比。此外，菲克定律 ${ }^{50}$ 是这些量之间的线性关系。最后， $\tau$ 与扩散时 间尺度有关。我们得出结论，LNET 成立。因此，Prigogine 为这个问题提出了一个MinEP 就不足为奇了 [19] – 见 Sect. 4.3.10。

## 物理代写|热力学代写Thermodynamics代考|Fick’s Law

$\frac{d a}{d t} \cdot d t=\left(\frac{\partial a}{\partial t}+\mathbf{v} \cdot \nabla a\right) \cdot d t=\left(\frac{\partial a}{\partial t}+\frac{d \mathbf{r}}{d t} \cdot \nabla a\right) \cdot d t$ 对于通用数量 $a$. 处于稳定状态 $\frac{\partial}{\partial t}=0$ ，然后 $d a=d \mathbf{r} \cdot \nabla a$. 让 $a=\mu_k$. 因此， $n_k d \mu_k=0$ 暗示 $\left(n_k \nabla \mu_k\right) \cdot d \mathbf{r}=0$ 对于任意 $d \mathbf{r}$ ， 因此:
$$0=n_k \nabla \mu_k=n_1 \nabla \mu_1+n_2 \nabla \mu_2$$

$$\sigma=-\mathbf{j} p k \cdot \nabla\left(\frac{\mu_k}{T}\right)=-\frac{\mathbf{j} p k}{T} \cdot \nabla \mu_k=-\frac{\mathbf{j} p 1}{T} \cdot \nabla \mu_1-\frac{\mathbf{j} p 2}{T} \cdot \nabla \mu_2=-\frac{1}{T}\left(\mathbf{j} p 1-\frac{n_1}{n_2} \mathbf{j} p 2\right) \cdot \nabla \mu_1$$

$v_k \nabla \cdot \mathbf{j}_k=\nabla \cdot\left(v_k \mathbf{j}_k\right)-\mathbf{j}_k \cdot \nabla v_k=\nabla \cdot\left(v_k \mathbf{j}_k\right)-\mathbf{j} k \cdot \nabla\left(\frac{\partial \mu_k}{\partial p}\right)_T$ 和
$\mathbf{j}_k \cdot \nabla\left(\frac{\partial \mu_k^0}{\partial p}\right)_T=\mathbf{j} p k \cdot \nabla\left(\frac{\partial \mu_k}{\partial p}\right) T=\left[\frac{\partial\left(\mathbf{j} p k \cdot \nabla \mu_k\right)}{\partial p}\right] T \propto\left[\frac{\partial\left(\mathbf{j} p k \cdot \nabla \mu_k\right)}{\partial p}\right] T=O\left(n_k \nabla \mu_k\right)=0$ 因为 $n_k \nabla \mu_k=0$. 因此，不可 压㿟性意味着 $\nabla \cdot\left(v_k \mathbf{j}_k\right)=0$. 连同定义 $\mathbf{j} p k \equiv \frac{\mathrm{j} k}{m_k}$ 和 $v_k^{\prime} \equiv m_k v_k$ ，这给出 ${ }^{53} \mathbf{j} p 1 v_1^{\prime}+\mathbf{j} p 2 v_2^{\prime}=0$. 然后
$$\sigma=-\frac{1}{T}\left(1+\frac{n_1 v_1^{\prime}}{n_2 v_2^{\prime}}\right) \mathbf{j} p 1 \cdot \nabla \mu_1=-\frac{1}{T}\left(1+\frac{n_1 v_1^{\prime}}{n_2 v_2^{\prime}}\right)\left(\frac{\partial \mu_1}{\partial n_1}\right) p, T \mathbf{j} p 1 \cdot \nabla n_1$$

## 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。