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物理代写|热力学代写Thermodynamics代考|Entropy Production in a Radiation Field

Just like Le Châtelier’s principle, the normalization constraint discussed in Sect. 6.2.5 holds regardless of the detailed nature of the heat transport mechanism. For example, radiative transfer carries energy through a succession of many events-including the emission, the transmission, the scattering and the absorption of photons. As a whole, however, the radiative transfer is an irreversible process, and it is perfectly meaningful to speak of the entropy density of radiation-for a thorough discussion, see Ref. [101]. ${ }^{45}$ In particular, when considering a set of atoms exchanging photons with each other, we may focus our attention on the set of photons with one particular frequency. With respect to this particular system, the ‘external world’ is made of both the atoms and all other photons. For the photons of our system, sending energy to the external world means just being absorbed by some atom, which will then interact with other atoms through other photons, etc. Analogously, our system receives energy from the external world when an atom emits a photon with the frequency of the photons of our system. The net energy flow to and from the external world depends on the state of the atoms, as the latter can emit and absorb photons. ${ }^{46}$ If the latter state is given, then our constrained minimization in Sect. 6.2.1 dictates that our system of photons has minimum $\overline{\int d \mathbf{x} \frac{P_h}{T}}$.

The physical meaning of this conclusion is clear if we recall that emission, scattering and absorption are the only physical processes occurring in our system, and that the only contribution to $P_h$ of these processes is of electromagnetic origin. After radiative transfer of heat has flattened all gradients of temperature while the photons reaching equilibrium with all the matter interacting with them, the only surviving contribution to the entropy production density $\sigma$ in the local entropy balance of Sect.4.2.8 is the electromagnetic one, which describes precisely the interaction of the electromagnetic field with matter. ${ }^{47}$ Thus, we conclude that the amount of entropy produced per unit time in our system of photons by emission and absorption tends to a minimum as the system relaxes to a stable, steady state. Our result is confirmed by the quantum mechanical analysis of $[103] .^{48}$

物理代写|热力学代写Thermodynamics代考|Uniform Temperature: A Reciprocal Problem,…

The limit of the vanishing temperature gradient deserves further discussion. This limit is justified whenever heating processes are not able to provide any significant temperature gradient across the system, due to competing energy transport. We limit ourselves to steady relaxed states for simplicity. The constrained minimization in Sect. 6.2.1 reduces to $\frac{1}{T} \int P_h d \mathbf{x}=\min$ with the constraint $\int P_h d \mathbf{x}=P_{T O T}$, which is equivalent to $T=\max$ with the constraint $\int P_h d \mathbf{x}=P_{T O T}$. Let the actual value of $T$ be $T_0$. Then, the reciprocity principle of Sect. $5.3 .12$ ensures that the solution of the latter variational problem is also the solution of the ‘reciprocal’ variational problem:
$$\int P_h d \mathbf{x}=\min ; \quad \text { with the constraint } \quad T=T_0$$

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。