### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT 40820

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Blackbody Radiation Distribution Function

By definition, a blackbody is a perfect absorber of thermal radiation; that is, it absorbs all incident radiation from all directions and at all wavelengths. It follows from this definition and from the Second Law of Thermodynamics that no body at a given temperature can emit more thermal radiation than a blackbody at the same temperature. Therefore, we say that a blackbody is an ideal emitter. Furthermore, it can be demonstrated through the use of “thought experiments” (see Ref. [1], pp. 32-33) that an isothermal enclosure is filled with blackbody radiation, which is both uniform and isotropic. According to the Stefan-Boltzmann law, the emissive power escaping from an isothermal enclosure through a vanishingly small hole in the wall is
$$e_{b}=\sigma T^{4}\left(\mathrm{~W} / \mathrm{m}^{2}\right),$$
where $T(\mathrm{~K})$ is the absolute temperature and $\sigma=5.6696 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \cdot \mathrm{K}^{4}$ is the Stefan-Boltzmann constant. The German physicist Josef Stefan (1835-1893) first suggested the form of Eq. (2.15) in 1879 on the basis of data already in the literature (see Problem 2.4) [2]. Stefan discovered that a straight line results when the initial cooling rate of a body suspended in a vacuum is plotted against the difference between its absolute temperature to the fourth power and that of its surroundings to the fourth power. Five years later Stefan’s student, Austrian physicist Ludwig Boltzmann (1844-1906), derived the form of Eq. (2.15) on the basis of classical thermodynamics [3]. Boltzmann’s derivation is also available in Ref. [1] (pp. $38-42$ ).

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Blackbody Properties

We learn in the first paragraph in the previous section that blackbody radiation is isotropic. We conclude that the intensity of a blackbody must be independent of direction; that is, for a blackbody
$$i_{\lambda}(\lambda, \vartheta, \varphi)=i_{b \lambda}(\lambda, T) .$$

It then follows from Eq. (2.11) that the directional spectral emissive power of a blackbody is
$$e_{b \lambda}(\lambda, T, \vartheta)=i_{b \lambda}(\lambda, T) \cos \vartheta .$$
Thus, the directional spectral emissive power of a blackbody varies as the cosine of the angle with respect to the local surface normal. This is sometimes referred to as Lambert’s cosine law, and surfaces that conform to this law are frequently referred to as Lambertian. While all blackbodies are Lambertian, not all Lambertian surfaces are blackbodies. The blackbody hemispherical spectral emissive power, $e_{b \lambda}(\lambda, T, \vartheta)$, is
$$e_{b \lambda}(\lambda, T)=i_{b \lambda}(\lambda, T) \int_{2 \pi} \cos \vartheta d \Omega=\pi i_{b \lambda}(\lambda, T),$$
and the blackbody total intensity, $i_{b}(T)$, is
$$i_{b}(T)=\int_{\lambda=0}^{\infty} i_{b \lambda}(\lambda, T) d \lambda .$$
Evaluation of the integral in Eq. (2.23) is complicated by the form of the integrand, given by Eq. (2.19). The approach is to introduce a change of variables, $\eta=C_{2} / \lambda T$, after which
$$i_{b}(T)=\frac{C_{1} T^{4}}{C_{2}^{4}} \int_{\eta=0}^{\infty} \frac{\eta^{3}}{e^{\eta}-1} d \eta .$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Emission and Absorption Mechanisms

To this point we have characterized thermal radiation as a wave phenomenon. However, in 1905 Albert Einstein introduced the idea of the photon as an alternative view of EM radiation. In Einstein’s photoelectric theory the photon is a particle whose energy $e$ (not to be confused with emissive power) is proportional to the frequency of a corresponding EM wave,
$$e=h v(\mathrm{~J}),$$
where $h$ is Planck’s constant. The dual wave-particle description of EM radiation is now firmly established, with one being more convenient to use than the other depending on the situation. Another important conclusion of the photoelectric theory is that, at the most fundamental level, radiation heat transfer always involves interactions between photons and electrons. Modern physics recognizes two categories of atomic particle: the fermions, which are the building blocks of matter, and bosons, which moderate interactions among the fermions. In the field of quantum electrodynamics (QED), photons and electrons form a boson-fermion pair whose interactions account for all electrical and magnetic phenomena. The reader interested in pursuing this fascinating topic further is referred to Richard P. Feynman’s highly readable classic QED: the Strange Theory of Light and Matter [8].

For the purposes of the following discussion, an atom may be viewed as a positively charged nucleus surrounded by a swarm of negatively charged electrons. In order for an atom to be electrically neutral, the number of electrons must exactly balance the positive charge of the nucleus. The rules of quantum mechanics require that the electrons organize themselves into layers, or “shells.” surrounding the nucleus. A discrete energy

level is identified with an electron depending on the shell it occupies, with electrons occupying the inner shells having less energy than those occupying outer shells. Electrons can migrate between shells only by gaining or giving up the amount of energy associated with the difference between their fixed energy in the two shells. The mechanism for gaining or giving up this energy is interaction with a photon, as required by Einstein’s photoelectric theory embodied in Eq. (2.27). Thus, when an electron moves from one energy level to another within an atom, the conservation of energy principle requires that a corresponding amount, or quantum, of energy be absorbed by or emitted from the atom. If the atomic transition occurs from energy level $E_{a}$ to a lower energy level $E_{b}$, then, according to Eq. (2.27), the frequency of the light emitted by the atom for this bound-bound transition is
$$v=\frac{E_{a}-E_{b}}{h} .$$

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