## 3 Numerical Results
### Gaussian Beam  
In the optical field, the Gaussian beam is a beam of monochromatic electromagnetic radiation which has a Gaussian intensity profile. As a result, it has a transverse electromagnetic field given by the Gaussian function. Such a beam can be expanded and focused through lens thus becomes different Gaussian beam at different time steps. According to the theory of quantum mechanics, electrons can both interact with other particles and be diffracted. It has properties of particle and wave [55]. For Gaussian beam case, a beam of electrons is a good model since both interaction between particles due to electromagnetic force and intensity of light should be simulated. In current paper, we choose an electron beam as model to be deformed by the electromagnet field. In addition, the external electric field plays a role of lens. In evolution of beam, the external electric field has an opposite direction to electric field induced by electron itself. In the Boltzmann equation, the acceleration term is from the electric field which includes two parts: Electric field \(\mathbf{E}_{self}\) induced by electron given by the Poisson's equation and external electric field \(\mathbf{E}_{appl}\). The external electric field is decided by self-consistent field method (SCF) using initial electric field induced by electrons. In current work, the external electric field is linear with respect to spatial coordinates. The relation between \(\mathbf{E}_{self}\) and \(\mathbf{E}_{appl}\) can be written as  
\[\mathbf{E}_{self}+\mathbf{E}_{appl}=-\omega^{2}\mathbf{r}, \tag{34}\]  
where \(\omega\) represents the difference between the initial self-consistent electric field due to particle charges and the linear external electric field. Its value is decided by a self-consistent domain. Figure 8: Development of the distribution function at \(x=0\) in nonlinear Landau damping case with non dimensional time. Figure 9: Development of entropy in nonlinear Landau damping case with non dimensional time. \[I=\left\{\left(\bar{x},\bar{y},\bar{u},\bar{v}\right)|\frac{\bar{x}^{2}}{a^{2}}+ \frac{\bar{y}^{2}}{\bar{a}^{2}}+\frac{\bar{u}^{2}}{\left(\omega a\right)^{2}}+ \frac{\bar{v}^{2}}{\left(\omega a\right)^{2}}=1\right\}. \tag{35}\]  
All values in Equation (35) are nondimensionalized by the root mean square (RMS) thermal velocity \(v_{th}\). The RMS results for space and particle velocity are computed by  
\[\sqrt{\bar{x}^{2}} =\sqrt{\frac{\int{x^{2}fd\[PERSON] d\Omega}}{\int{fd\[PERSON] d\Omega}}}=a, \sqrt{\bar{y}^{2}}=\sqrt{\frac{\int{y^{2}fd\[PERSON] d\Omega}}{\int{fd\[PERSON] d\Omega}}}=a, \tag{36}\] \[\sqrt{\bar{u}^{2}} =\sqrt{\frac{\int{u^{2}fd\[PERSON] d\Omega}}{\int{fd\[PERSON] d\Omega}}}= \omega a,\sqrt{\bar{v}^{2}}=\sqrt{\frac{\int{v^{2}fd\Xi d\Omega}}{\int{fd\Xi d \Omega}}}=\omega a.\]  
For Gaussian beam case, the initial self-consistent electric field is linear in space. So we can obtain the two unknown variables \(\omega\) and \(a\) using RMS thermal velocity \(v_{th}\),  
\[a=\frac{r_{a}}{2},v_{th}=\frac{\omega r_{a}}{2}, \tag{37}\]  
where \(r_{a}\) is the radius of beam. Now we introduce a tune depression factor \(\chi=\frac{\omega_{0}}{\omega}\) such that we can adjust external electric field with initial self-consistent electric field. The linear external electric field whose direction is opposite to initial self-consistent electric field can be written as  
\[\mathbf{E}_{appl}=\omega_{0}^{2}\mathbf{r}. \tag{38}\]  
In the evolution of a beam, the external electric field \(\mathbf{E}_{appl}\) is used to focalize the beam. The dimensionless form of Equation (1) can be rewritten as  
\[\frac{\partial f}{\partial t}+\mathbf{u}\frac{\partial f}{\partial x}+\left(- \left(\mathbf{E}_{self}+\mathbf{E}_{app}\right)\right)\frac{\partial f}{\partial\mathbf{ u}}=\frac{g-f}{\tau}. \tag{39}\]  
The negative sign \" - \" is because of the negative charge carried by electron. The 2D initial condition is written as  
\[f_{0}=\left\{