2024.05.25阶段性总结

最近一段时间的主要工作：
1.专业课学习（电动力学，量子力学，统计物理）；
2.熟悉srt需要用到的知识，包括激光等离子体加速物理，同时也需要学习一定程度的python（计划下学期或者下下学期选修计算物理，量子束流物理与能源物理基础）；
3.读文献（srt\论文中的所有文献），重点是综述；
4.准备六级考试（06.15），并且暑假如果有时间可以准备并考一次雅思；
5.假期要学习并掌握Smilei-PIC与FLASH流体模拟的使用（如果有空闲，也可以试试Multi）；
6.准备中世纪西方政治（06.12）和马原的结课论文，马原需要开始准备考试内容了。

目前已完成的进度:
电动力学复习了电磁场的基本性质与相对论电动力学；初步理解了激光等离子体物理相关的一些知识，林老师的出站论文看了一部分；基本了解了python的语法，可以较为熟练地使用GitHub，cmd以及终端进行操作；在Smilei上初步实现了一些模拟，了解了基本的操作，FLASH已经安装。

FLASH官网地址：FLASH
Smilei-PIC官网地址：Smilei-PIC
四六级考试官网地址：四六级

Chapter 1 Basic Equations of Vacuum Electromagnetic Field

Before the birth of electrodynamics, people had already summarized a large number of experimental laws on electromagnetic phenomena, including but not limited to the following:
1.Coulomb’s law：

$$\vec{F}_{12}=\frac{Q_1Q_2}{4\pi \varepsilon_0 \left | \vec{r}_{12} \right |^3 }\hat{r}_{12},$$

which describe the electrostatic interaction between two point charges;
We can define the concept of electric field:

$$\vec{E}(\vec{x})=\frac{1}{4\pi \varepsilon _0} \int_{V^{\prime }} \rho (\vec{x^{\prime }} )\frac{\vec{r} }{r^3} \mathrm{d}V^{\prime },\ \mathrm{d}\vec{F}(\vec{x} )=\rho (\vec{x} )\vec{E}(\vec{x} )\mathrm{d}V$$

This definition reflects the nature of electrical interaction: there is no direct interaction between charges, and the transfer of their energy and momentum depends on the electric field - the source charge excites the electric field, and then the electric field exerts a force on other charges. This also seems to explain one thing: the field itself also has a certain momentum and energy, which is similar to the “matter” in our usual sense.
2.Biot-Savart law:

$$\vec{F}_{12}=\frac{\mu _0I_1I_2}{4\pi } \oint_{l_1 } \oint_{l_2}\frac{\mathrm{d}\vec{l}_2 \times (\mathrm{d}\vec{l}_1 \times \hat{r}_{12} )}{r_{12}^2} ,$$

which describes the total magnetic interaction between two coils carrying a steady current.Similarly, this equation can also be written in the form of field interaction:

$$\vec{B}(\vec{x})=\frac{\mu _0}{4\pi } \int_{V^{\prime }} \vec{J} (\vec{x^{\prime }} )\times\frac{\vec{r} }{r^3} \mathrm{d}V^{\prime },\ \mathrm{d}\vec{F}(\vec{x} )=\vec{J} (\vec{x} )\times \vec{B}(\vec{x} )\mathrm{d}V$$

3.Law of Charge Conservation:
The change in the amount of charge in any region of space is equal to the amount of charge flowing into the region minus the amount of charge flowing out of the region.We can express them using a continuity equation for current and charge:

$$\nabla \cdot \vec{J}+\frac{\partial \rho }{\partial t}=0$$

4.Faraday’s Law:
When the magnetic flux through the coil changes, an induced electromotive force is generated in the coil. The direction of the induced electromotive force is determined by Lenz’s law: the magnetic field generated by the induced current in the coil compensates for the change in magnetic flux.

$$\oint \vec{E}\cdot d\vec{l}=-\frac{\mathrm{d} }{\mathrm{d} t} \int_{S}\vec{B}\cdot d\vec{S}$$

Don’t forget that the essence of electric current is the flow of electric charges, so we can ask a question: since electric charges generate electric fields and electric current generates magnetic fields, can the moving electric field directly generate a magnetic field?
Faraday’s law tells us that a changing magnetic field will generate an electric field (this electric field is slightly different from a general static electric field, for example, it has no “source”). If our conjecture is correct, a moving electric field, or a changing electric field, can also generate a magnetic field. Then, are electrical and magnetic phenomena essentially the same?
These questions were eventually answered by the brilliant physicist Maxwell.This genius physicist boldly took the properties of the field summarized from the experimental laws as universal laws, and combined the law of conservation of charge and symmetry to propose the displacement current hypothesis. He summarized them all into four clear and profound equations - Maxwell’s equations:

$$\begin{cases}\nabla \times \vec{E}=-\frac{\partial \vec{B} }{\partial t}, \\\nabla \cdot \vec{E}=\frac{\rho}{\varepsilon_0}, \\\nabla \times \vec{B}=\mu _0\vec{J}+\mu_0\varepsilon_0 \frac{\partial \vec{E} }{\partial t} ,\\\nabla \cdot \vec{B}=0,\ \end{cases}$$

Maxwell’s equations unified electricity and magnetism in an extremely symmetrical mathematical form. In this process, he made a lot of generalizations, which needed to be verified in the experimental area.The two equations related to curl in the equation group describe the phenomenon of mutual excitation between electricity and magnetism. This alternating excitation process is easily reminiscent of “waves”.Maxwell then boldly stated the existence of “electromagnetic waves” and further predicted that light was a kind of electromagnetic wave.This was proved by Hertz in the laboratory after his death.This established Maxwell’s position in physics and also brought a new era to the development of human civilization.
We can rewrite the four first-order differential equations in Maxwell’s equations into two second-order differential equations, which describe the space-time distribution of the electric field and magnetic field respectively:

$$\begin{cases}\nabla ^2\vec{E}-\mu _0\varepsilon_0\frac{\partial^2 \vec{E} }{\partial t^2} =\nabla(\frac{\rho}{\varepsilon_0} )+\frac{\partial}{\partial t}(\mu _0\vec{J} ),\\ \nabla ^2\vec{B}-\mu _0\varepsilon_0\frac{\partial^2 \vec{B} }{\partial t^2} =-\nabla \times (\mu _0\vec{J} ),\ \end{cases}$$

It is easy to see that these are two traveling wave equations,wave speed is:

$$c=\frac{1}{\sqrt{\mu _0\epsilon _0} }$$

Based on the measurement results of vacuum dielectric constant and vacuum magnetic permeability, the speed of vacuum electromagnetic waves can be calculated, which is exactly the speed of vacuum light.This is one of the reasons why Maxwell made this judgment.
Now we have the basic equations for the electromagnetic field, but this is not enough. We also need to describe how the field acts on charged bodies.Let’s go back to the Coulomb’s law and the Biot-Savart law in the field form we gave at the beginning.We consider the charged body under the action of the field as a point charge moving at a certain speed:

$$\vec{F} =\int_{V^{\prime }} (\rho \vec{E}+\vec{J}\times \vec{B})\mathrm{d}V^{\prime }\Rightarrow \vec{f}=\rho \vec{E}+\vec{J}\times \vec{B},$$ $$\rho (\vec{x^{\prime }},t )=q\delta (\vec{x^{\prime }}-\vec{x}_q(t) ) ,\vec{J}(\vec{x^{\prime }},t )=\rho \vec{v}=q\vec{v} \delta (\vec{x^{\prime }}-\vec{x}_q(t) ) ,$$

Finally we get the expression of the Lorentz force:

$$\vec{F}=q(\vec{E}+\vec{v}\times \vec{B} ),$$

We mentioned earlier that electromagnetic fields are also substances with energy and momentum.Now, starting from the Lorentz force, we can study the energy and momentum exchange between the electromagnetic field and mechanical objects, and then give expressions for the energy and momentum of the electromagnetic field.
We consider the rate of change of the total mechanical energy in a volume element.According to the kinetic energy theorem:

$$\frac{\partial w_{src}}{\partial t}=\vec{f}\cdot \vec{v}= (\rho \vec{E}+\vec{J}\times \vec{B}) \cdot \vec{v}=\vec{E}\cdot \vec{J}$$

Now, we use Maxwell’s equations and replace the current with the field strength. After some mathematical manipulation, we can get the result in the following form：

$$\frac{\partial w_{src}}{\partial t}=-\frac{\partial u}{\partial t}-\nabla \cdot \vec{S} ,\ u=\frac{B^2}{2\mu _0}+\frac{\varepsilon_0E^2}{2}, \ \vec{S}=\frac{1}{\mu _0}\vec{E}\times \vec{B},$$

This is obviously a continuity equation, which we can write as follow:

$$\frac{\partial (w_{src}+u)}{\partial t}+\nabla \cdot \vec{S}=0 ,$$

Therefore, we define “$u$” as the energy density of the electromagnetic field and “$\vec{S} $” as the energy flow density of the electromagnetic field.For momentum, we adopt the same approach and express the Lorentz force density in terms of field strength:

$$\vec{f}=-(\nabla u+\frac{1}{c^2}\frac{\partial\vec{S} }{\partial t} ) + \textstyle \sum_{ij}\partial _j(\varepsilon_0E_iE_j+\frac{1}{\mu _0}B_iB_j )\hat{e}_i$$

We would like to write the momentum theorem as a continuity equation:

$$\frac{\partial }{\partial t }\vec{p}_{src} = \vec{f}=-\frac{\partial\vec{g}}{\partial t} -\nabla\cdot \overleftrightarrow{T},\ \vec{g}=\frac{\vec{S} }{c^2} ,\ \overleftrightarrow{T} =\varepsilon\vec{E}\vec{E} +\frac{1}{\mu _0}\vec{B}\vec{B}-u\overleftrightarrow{I} ,$$

Similarly, “$\vec{g} $” is considered as the momentum density of the electromagnetic field, and “$\overleftrightarrow{T} $” is considered as the momentum current density tensor or Maxwell stress tensor of the electromagnetic field.
Now that we know the basic equations of motion of electromagnetic fields, electromagnetic field energy and momentum, and the expression of the interaction between electromagnetic fields and charged bodies, we will go a step further and explore how electromagnetic fields are generated and propagated.

Chapter 2 Excitation of Electromagnetic Wave

3.电子加热机制：

3.4$J\times B$加热：

3.电子加热机制：

3.3真空加热：

3.电子加热机制：

3.2共振吸收：

3.电子加热机制：

3.1碰撞吸收：

2.激光与等离子体相互作用：

2.4相对论激光的相关性质：

2.激光与等离子体相互作用：

2.3激光在等离子体中的传播：

2.激光与等离子体相互作用：

2.2等离子体双流体模型：

2.激光与等离子体相互作用：

2.1等离子体内部的屏蔽效应：