Lorentz collision term  洛伦兹碰撞项

Chao Dong ${}^1$${}^1$^(1){ }^{1}${}^1$ 董超 ${}^1$${}^1$^(1){ }^{1}${}^1$

Beijing National Laboratory for Condensed Matter Physics and Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
中国科学院物理研究所北京凝聚态物理国家实验室和软物质物理实验室, 北京 100190

(*Electronic mail: chaodong @iphy.ac.cn)
（*邮箱：chaodong@iphy.ac.cn）

(Dated: 3 June 2024)
（日期：2024 年 6 月 3 日）

The electron Fokker-Planck collision term is
电子福克-普朗克碰撞项是

where $f_e$$f_e$f_(e)f_{e}$f_e$ is the electron distribution function, $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ is the dynamical friction coefficient, and $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ is the diffusion coefficient.
其中 $f_e$$f_e$f_(e)f_{e}$f_e$ 是电子分布函数， $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ 是动摩擦系数， $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ 是扩散系数。

In the Lorentz model, only electron-ion collisions are considered in the electron collision term, in which the ions are taken to be stationary. In the binary collision theory, $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ and $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ result from the successive binary collisions and are given by
在洛伦兹模型中，电子碰撞项仅考虑电子-离子碰撞，其中离子被视为静止。在二元碰撞理论中， $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ 和 $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ 由连续的二元碰撞产生，由下式给出

where $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ is the electron velocity change during a binary collision specified by a vector $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ with the ion. The length of $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ is the impact parameter $b$$b$bb$b$ which is the distance of closest approach between the electron and ion when their mutual interaction is not considered, and its direction is pointing from the ion to the electron when the distance of closest approach is reached. ${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$d^(2)b=bdbdphi\mathrm{d}^{2} \mathbf{b}=b \mathrm{~d} b \mathrm{~d} \phi${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$ with $\varphi$$\varphi$phi\phi$\varphi$ being the angle between $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ and a fixed plane containing ${\mathbf{v}}_e$${\mathbf{v}}_e$v_(e)\mathbf{v}_{e}${\mathbf{v}}_e$. Expressed in terms of the scattering angle $\theta$$\theta$theta\theta$\theta$ for ${\mathbf{v}}_e,\mathrm{\Delta }{\mathbf{v}}_e$${\mathbf{v}}_e,\mathrm{\Delta }{\mathbf{v}}_e$v_(e),Deltav_(e)\mathbf{v}_{e}, \Delta \mathbf{v}_{e}${\mathbf{v}}_e,\mathrm{\Delta }{\mathbf{v}}_e$ is given by
其中 $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ 是由向量 $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ 指定的电子与离子二元碰撞期间电子的速度变化。 $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ 的长度即为撞击参数 $b$$b$bb$b$ ，即在不考虑电子与离子相互作用的情况下，电子与离子最接近的距离，其方向为当电子离子达到最近距离时，从离子指向电子。 ${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$d^(2)b=bdbdphi\mathrm{d}^{2} \mathbf{b}=b \mathrm{~d} b \mathrm{~d} \phi${\mathrm{d}}^2\mathbf{b}=b\mathrm{d}b\mathrm{d}\varphi$ 其中 $\varphi$$\varphi$phi\phi$\varphi$ 是 $\mathbf{b}$$\mathbf{b}$b\mathbf{b}$\mathbf{b}$ 与包含 ${\mathbf{v}}_e$${\mathbf{v}}_e$v_(e)\mathbf{v}_{e}${\mathbf{v}}_e$ 的固定平面之间的角度。以ve的散射角 $\theta$$\theta$theta\theta$\theta$ 表示，Δve由下式给出

where $\hat{\mathbf{b}}\equiv \mathbf{b}/b$$\widehat{\mathbf{b}}\equiv \mathbf{b}/b$hat(b)-=b//b\hat{\mathbf{b}} \equiv \mathbf{b} / b$\hat{\mathbf{b}}\equiv \mathbf{b}/b$. Using the Rutherford scattering formula
其中 $\hat{\mathbf{b}}\equiv \mathbf{b}/b$$\widehat{\mathbf{b}}\equiv \mathbf{b}/b$hat(b)-=b//b\hat{\mathbf{b}} \equiv \mathbf{b} / b$\hat{\mathbf{b}}\equiv \mathbf{b}/b$ 。使用卢瑟福散射公式

for the bare Coulomb interaction, where $b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$$b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$b_(0)-=Ze^(2)//(4piepsi_(0)m_(e)v_(e)^(2))b_{0} \equiv Z e^{2} /\left(4 \pi \varepsilon_{0} m_{e} v_{e}^{2}\right)$b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$ is the impact parameter for which $\theta ={90}^{\circ }$$\theta ={90}^{\circ }$theta=90^(@)\theta=90^{\circ}$\theta ={90}^{\circ }$ with $Z$$Z$ZZ$Z$ being the ion charge number, $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ can be expressed as
对于裸库仑相互作用，其中 $b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$$b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$b_(0)-=Ze^(2)//(4piepsi_(0)m_(e)v_(e)^(2))b_{0} \equiv Z e^{2} /\left(4 \pi \varepsilon_{0} m_{e} v_{e}^{2}\right)$b_0\equiv Z{e}^2/\left(4\pi {\epsilon }_0{m}_e{v}_e^2\right)$ 是冲击参数 (当 $\theta ={90}^{\circ }$$\theta ={90}^{\circ }$theta=90^(@)\theta=90^{\circ}$\theta ={90}^{\circ }$ 取b0) ， $Z$$Z$ZZ$Z$ 是离子电荷数， $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ 可以是表示为

Substituting the above expression for $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ into Eqs. (2) and (3) and carrying out the integrals over $\varphi$$\varphi$phi\phi$\varphi$ gives
将上面的 $\mathrm{\Delta }{\mathbf{v}}_e$$\mathrm{\Delta }{\mathbf{v}}_e$Deltav_(e)\Delta \mathbf{v}_{e}$\mathrm{\Delta }{\mathbf{v}}_e$ 表达式代入等式 (2) 和 (3) 对 $\varphi$$\varphi$phi\phi$\varphi$ 进行积分得到

where I is the unit dyadic. The integrals over $b$$b$bb$b$ in the above two equations are divergent as $b\to \mathrm{\infty }$$b\to \mathrm{\infty }$b rarr oob \rightarrow \infty$b\to \mathrm{\infty }$. The divergence arises from the neglect of collective interactions in the binary collision theory. To cure the divergence, an upper cutoff at the Debye length ${\lambda }_D$${\lambda }_D$lambda_(D)\lambda_{D}${\lambda }_D$ is introduced for $b$$b$bb$b$. Carrying out the integrals over $b$$b$bb$b$ in Eqs (7) and (8) thus gives
其中 I 是单位并矢。上述两个方程中 $b$$b$bb$b$ 上的积分发散为 $b\to \mathrm{\infty }$$b\to \mathrm{\infty }$b rarr oob \rightarrow \infty$b\to \mathrm{\infty }$ 。这种分歧源于二元碰撞理论中对集体相互作用的忽视。为了解决发散问题，为 $b$$b$bb$b$ 引入了德拜长度 ${\lambda }_D$${\lambda }_D$lambda_(D)\lambda_{D}${\lambda }_D$ 处的上限截止。对式(7)和(8)中的 $b$$b$bb$b$ 进行积分，得到

For the weakly coupled plasmas, the ration of ${\lambda }_D$${\lambda }_D$lambda_(D)\lambda_{D}${\lambda }_D$ to the average value of $b_0$$b_0$b_(0)b_{0}$b_0$ is a very large number in the sense that its logarithm is much larger than 1 . Within the logarithmic accuracy, the terms of the order of unity can be neglected compared to the big logarithm and the weak dependence of the logarithm on $v_e$$v_e$v_(e)v_{e}$v_e$ can be eliminated by choosing $v_e\sim v_{\text{the}}$$v_e\sim v_{\text{the }}$v_(e)∼v_("the ")v_{e} \sim v_{\text {the }}$v_e\sim v_{\text{the }}$ where $v_{\text{the}}\equiv \sqrt{k_B{T}_e/m_e}$$v_{\text{the }}\equiv \sqrt{k_B{T}_e/m_e}$v_("the ")-=sqrt(k_(B)T_(e)//m_(e))v_{\text {the }} \equiv \sqrt{k_{B} T_{e} / m_{e}}$v_{\text{the }}\equiv \sqrt{k_B{T}_e/m_e}$ is the electron thermal velocity. In this way, $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ and $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ are simplified to
对于弱耦合等离子体， ${\lambda }_D$${\lambda }_D$lambda_(D)\lambda_{D}${\lambda }_D$ 与 $b_0$$b_0$b_(0)b_{0}$b_0$ 平均值之比是一个非常大的数，因为它的对数远大于1。在对数精度内，与大对数相比，可以忽略单位阶项，并且可以通过选择 $v_e\sim v_{\text{the}}$$v_e\sim v_{\text{the }}$v_(e)∼v_("the ")v_{e} \sim v_{\text {the }}$v_e\sim v_{\text{the }}$ 来消除对数对 $v_e$$v_e$v_(e)v_{e}$v_e$ 的弱依赖性，其中 $v_{\text{the}}\equiv \sqrt{k_B{T}_e/m_e}$$v_{\text{the }}\equiv \sqrt{k_B{T}_e/m_e}$v_("the ")-=sqrt(k_(B)T_(e)//m_(e))v_{\text {the }} \equiv \sqrt{k_{B} T_{e} / m_{e}}$v_{\text{the }}\equiv \sqrt{k_B{T}_e/m_e}$ 是电子热速度。这样， $⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e):)\left\langle\Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e⟩$ 和 $⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$(:Deltav_(e)Deltav_(e):)\left\langle\Delta \mathbf{v}_{e} \Delta \mathbf{v}_{e}\right\rangle$⟨\mathrm{\Delta }{\mathbf{v}}_e\mathrm{\Delta }{\mathbf{v}}_e⟩$ 就简化为

where $\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$$\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$ln Lambda=ln(lambda_(D)//lambda_(L))\ln \Lambda=\ln \left(\lambda_{D} / \lambda_{L}\right)$\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$ is the Coulomb logarithm and ${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$lambda_(L)=Ze^(2)//(4piepsi_(0)k_(B)T_(e))\lambda_{L}=Z e^{2} /\left(4 \pi \varepsilon_{0} k_{B} T_{e}\right)${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$ is the Landau length. Substituting Eqs. (11) and (12) into Eq. (1) and using
其中 $\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$$\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$ln Lambda=ln(lambda_(D)//lambda_(L))\ln \Lambda=\ln \left(\lambda_{D} / \lambda_{L}\right)$\ln \mathrm{\Lambda }=\ln \left({\lambda }_D/{\lambda }_L\right)$ 是库仑对数， ${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$lambda_(L)=Ze^(2)//(4piepsi_(0)k_(B)T_(e))\lambda_{L}=Z e^{2} /\left(4 \pi \varepsilon_{0} k_{B} T_{e}\right)${\lambda }_L=Z{e}^2/\left(4\pi {\epsilon }_0{k}_B{T}_e\right)$ 是朗道长度。代入等式。将（11）和（12）代入等式 (1) 并使用

we obtain 我们获得

This is the Lorentz collision term ${}^1$${}^1$^(1){ }^{1}${}^1$.
这是洛伦兹碰撞项 ${}^1$${}^1$^(1){ }^{1}${}^1$ 。

${}^1$${}^1$^(1){ }^{1}${}^1$ R. J. Goldston and P. H. Rutherford, Introduction to plasma Physics (Institute of Physics, Bristol, 1995), Chap. 13.
${}^1$${}^1$^(1){ }^{1}${}^1$ R. J. Goldston 和 P. H. Rutherford，等离子体物理学简介（布里斯托尔物理研究所，1995 年），第 13 章。