⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 075201 Electron-Electron Collision Term Describing the Reflections Induced Scattering in a Magnetized Plasma * Chao Dong (董超)1,2**, Ding Li (李定)1,3,2**, Chang Jiang (姜畅)1,2 Affiliations 1Beijing National Laboratory for Condensed Matter Physics and CAS Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 2University of Chinese Academy of Sciences, Beijing 100049 3Songshan Lake Materials Laboratory, Dongguan 523808 Received 2 February 2019, online 20 June 2019 *Supported by the National MCF Energy R&D Program under Grant No 2018YFE0311300, the National Natural Science Foundation of China under Grant Nos 11875067, 11835016, 11705275, 11675257 and 11675256, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No XDB16010300, the Key Research Program of Frontier Science of Chinese Academy of Sciences under Grant No QYZDJ-SSW-SYS016, and the External Cooperation Program of Chinese Academy of Sciences under Grant No 112111KYSB20160039.
**Corresponding author. Email: chaodong@iphy.ac.cn; dli@iphy.ac.cn Citation Text: Dong C, Li D and Jiang C 2019 Chin. Phys. Lett. 36 075201    Abstract For an electron-electron collision with characteristic scale length larger than the relative gyro-radius of the two colliding electrons, when the initial relative parallel kinetic energy cannot surmount the Coulomb repulsive potential, reflection will occur with interchange of the parallel velocities of the two electrons after the collision. The Fokker–Planck approach is employed to derive the electron collision term $\mathcal{C}_{\rm R}$ describing parallel velocity scattering due to the reflections for a magnetized plasma where the average electron gyro-radius is much smaller than the Debye length but much larger than the Landau length. The electron parallel velocity friction and diffusion coefficients due to the reflections are evaluated, which are found not to depend on the electron perpendicular velocity. By studying the temporal evolution of the $H$ quantity due to $\mathcal{C}_{\rm R}$, it is found that $\mathcal{C}_{\rm R}$ eventually makes the system relax to a state in which the electron parallel velocity distribution is decoupled from the perpendicular velocity distribution. DOI:10.1088/0256-307X/36/7/075201 PACS:52.20.Fs, 52.25.Dg, 52.25.Xz © 2019 Chinese Physics Society Article Text In today's large tokamaks, there usually exists such a strong magnetic field that the electron gyro-radius $\rho_{\rm e}=\bar{v}/{\it \Omega}$ is smaller than the Debye length $\lambda_{\rm D}=\sqrt{\varepsilon_0 k_{_{\rm B}} T/(ne^2)}$ but much larger than the Landau length $\lambda_{\rm L}=e^2/(4\pi\varepsilon_0 k_{_{\rm B}} T)$ for the edge plasmas, where $\bar{v}=\sqrt{k_{_{\rm B}} T/m}$ is the electron rms velocity, ${\it \Omega}=eB/m$ is the electron gyro-frequency, $B$ is the magnitude of the magnetic field, $\varepsilon_0$ is the permittivity of the vacuum, $k_{_{\rm B}}$ is the Boltzmann constant, and $-e$, $m$, $n$, and $T$ are the electron charge, mass, number density, and temperature, respectively. In this case, the magnetic field effects on the electron involved collisions with characteristic scale length $L_{\rm c}$ larger than $\rho_{\rm e}$ have to be taken into account. The seemingly simple binary Coulomb collision problem in a magnetic field is in fact a problem of considerable complexity and was frequently treated by regarding the Coulomb interaction as a perturbation to the particles' helical motions.[1–4] This perturbative treatment fails when the relative motion of the two colliding particles is changed notably by the interaction, as the collision process is determined by the relative coordinates. For an electron-electron (e–e) collision with $L_{\rm c}$ larger than $\rho_{e}$, the collision process is approximately one-dimensional and the total perpendicular kinetic energy of the two colliding electrons is an adiabatic invariant.[5,6] Perpendicular and parallel refer to directions relative to the magnetic field. In this case, the magnetic field together with the interaction potential forms a potential barrier. When the kinetic energy of the relative parallel motion before the collision is smaller than the energy of the potential barrier, reflection with interchange of the parallel velocities of the two electrons after the collision will occur. For repulsive interactions, reflections in binary collisions have been investigated theoretically[2,7,8] and numerically.[9] Once a reflection occurs, it is obviously inappropriate to adopt the perturbation method to study the collision process, which implies that the magnetized collision terms[10–17] derived based on the perturbation approximation do not take the e–e collisions with reflections into account properly. For $\rho_{\rm e} \ll \lambda_{\rm L}$, O'Neil[5] derived a Boltzmann-like collision term from the BBGKY hierarchy to describe the e–e collisions after a close analytical investigation of the collision process. He demonstrated that the distant e–e collisions with reflections dominated the scattering of the electron parallel velocity. Psimopoulos and Li[7] constructed a nonlocal Boltzmann-like collision term due to reflections in like-particle collisions for a plasma in an infinitely strong magnetic field. They found that reflections could serve as a mechanism of heat transport across the magnetic field when the nonlocality of the collision processes was considered, and estimated the corresponding thermal diffusivity which was shown to result primarily from the distant collisions. Strictly speaking, the Boltzmann collision term is not suited to describe the distant collisions with $L_{\rm c}$ larger than the interparticle spacing, in which many particles interact simultaneously and weakly. The Fokker–Planck collision term is tailored for this scenario.[18–20] Dubin[8] employed the Fokker–Planck approach to study the parallel velocity scattering for both attractive and repulsive interactions in a magnetized plasma where the particles' gyro-radii were smaller than $\lambda_{\rm D}$. The influence on the collision process of the parallel velocity diffusion due to the interactions of the colliding pair with surrounding charges has been considered and many impressive results were achieved. At the same time, Dubin's research was restricted to a Maxwellian relative parallel velocity distribution. Its generalization to non-Maxwellian distributions is needed. In this Letter, we consider the electron parallel velocity scattering due to e–e collisions based on the Fokker–Planck approach for a magnetized plasma where $\lambda_{\rm L} \ll \rho_{\rm e} < \lambda_{\rm D}$. Different from Dubin's work,[8] the relative velocity distribution is not assumed to be Maxwellian and the parallel velocity diffusion influence on the collision process is ignored for simplicity. For e–e collisions with $L_{\rm c} < \rho_{\rm e}$, the magnetic field effects are trivial. On this account, our attention will be focused on the e–e collisions with $L_{\rm c}>\rho_{\rm e}$ in what follows. Considering the electrostatic interaction of two electrons in the presence of a strong and uniform magnetic field, ${\boldsymbol B}=B\hat{\boldsymbol e}_z$, the motion equations for the two electrons are $$\begin{align} m\frac{d{\boldsymbol v}_1}{dt}=\,&-e{\boldsymbol v}_1\times{\boldsymbol B}+\frac{e^2}{4\pi\varepsilon_0}\frac{{\boldsymbol r}_1-{\boldsymbol r}_2}{|{\boldsymbol r}_1-{\boldsymbol r}_2|^3},~~ \tag {1} \end{align} $$ $$\begin{align} m\frac{d{\boldsymbol v}_2}{dt}=\,&-e{\boldsymbol v}_2\times{\boldsymbol B}+\frac{e^2}{4\pi\varepsilon_0}\frac{{\boldsymbol r}_2-{\boldsymbol r}_1}{|{\boldsymbol r}_1-{\boldsymbol r}_2|^3},~~ \tag {2} \end{align} $$ where ${\boldsymbol r}_i$ and ${\boldsymbol v}_i$ are the coordinate and velocity of electron $i (i=1,2)$, respectively. Equation (2) minus Eq. (1) gives $$\begin{align} m\frac{d{\boldsymbol v}}{dt}=-e{\boldsymbol v}\times{\boldsymbol B}+\frac{2e^2}{4\pi\varepsilon_0} \frac{{\boldsymbol r}}{|{\boldsymbol r}|^3},~~ \tag {3} \end{align} $$ where ${\boldsymbol r}={\boldsymbol r}_2-{\boldsymbol r}_1$ is the coordinate of electron 2 relative to that of electron 1, and ${\boldsymbol v}=d{\boldsymbol r}/dt$ is the relative velocity. Making guiding center transformation through $$\begin{align} {\boldsymbol R}={\boldsymbol r}-{\boldsymbol \rho},~~ \tag {4} \end{align} $$ where ${\boldsymbol \rho}={\boldsymbol v}\times\hat{\boldsymbol e}_z/{\it \Omega}$ is the relative gyro-radius, we find that $$\begin{align} \frac{d{\boldsymbol R}_\perp}{dt}=-\frac{2e}{4\pi\varepsilon_0 B}\frac{({\boldsymbol R}_\perp+{\boldsymbol \rho})\times\hat{\boldsymbol e}_z}{[|{\boldsymbol R}_\perp+{\boldsymbol \rho}|^2+z^2]^{3/2}},~~ \tag {5} \end{align} $$ where $z$ and ${\boldsymbol R}_\perp$ are the parallel and perpendicular components of ${\boldsymbol R}$, respectively. For the relative motion in the $z$-direction, we have $$\begin{align} \frac{dz}{dt}=\,&v_z,~~ \tag {6} \end{align} $$ $$\begin{align} \frac{dv_z}{dt}=\,&\frac{e^2}{4\pi\varepsilon_0 \mu}\frac{z}{[|{\boldsymbol R}_\perp+{\boldsymbol \rho}|^2+z^2]^{3/2}},~~ \tag {7} \end{align} $$ where $\mu=m/2$ is the reduced mass, and $v_z=v_{2z}-v_{1z}$ is the parallel component of ${\boldsymbol v}$ with $v_{1z}$ and $v_{2z}$ being the parallel components of ${\boldsymbol v}_1$ and ${\boldsymbol v}_2$, respectively. As we consider only those e–e collisions with $L_{\rm c}>\rho=|{\boldsymbol \rho}|$, the guiding center approximation can be used. Equations (5) and (7) thus become $$\begin{align} \frac{d{\boldsymbol R}_\perp}{dt}=\,&-\frac{2e}{4\pi\varepsilon_0 B}\frac{{\boldsymbol R}_\perp\times\hat{\boldsymbol e}_z}{(|{\boldsymbol R}_\perp|^2+z^2)^{3/2}},~~ \tag {8} \end{align} $$ $$\begin{align} \frac{dv_z}{dt}=\,&\frac{e^2}{4\pi\varepsilon_0 \mu}\frac{z}{(|{\boldsymbol R}_\perp|^2+z^2)^{3/2}}.~~ \tag {9} \end{align} $$ From Eq. (8), it follows that $$\begin{align} |{\boldsymbol R}_\perp|={\rm const}=p,~~ \tag {10} \end{align} $$ where $p$ can be regarded as the impact parameter equivalent to that in the unmagnetized case. Substituting the above equation into Eq. (9), multiplying $v_z$ on both sides, and integrating over $t$, we obtain $$\begin{align} \frac{1}{2}\mu v_z^2+\frac{e^2}{4\pi\varepsilon_0(p^2+z^2)^{1/2}}=\frac{1}{2}\mu v_{z0}^2,~~ \tag {11} \end{align} $$ where $v_{z0}=v_z(t=-\infty)$ is the initial relative parallel velocity. After the collision, $|z|\to \infty$ and the potential energy in Eq. (11) tends to zero, thus $$\begin{align} v_z^2(t=\infty)=v_{z0}^2.~~ \tag {12} \end{align} $$ Two cases can be distinguished. If $$\begin{align} v_z(t=\infty)=v_{z0},~~ \tag {13} \end{align} $$ the two electrons will cross each other and retain their initial parallel velocities after the collision. In this case, no momentum and energy transfer between the two electrons. If $$\begin{align} v_z(t=\infty)=-v_{z0},~~ \tag {14} \end{align} $$ reflection occurs and the two electrons interchange their initial parallel velocities after the collision. In this case, the parallel velocity change of electron 1 through collision with electron 2 is $$\begin{alignat}{1} \!\!\!\!\!\!\!\Delta v_{1z}=v_{1z}(t=\infty)-v_{1z}(t=-\infty)=v_{2z}-v_{1z}.~~ \tag {15} \end{alignat} $$ In the second step of the above equation, $v_{1z}(t=-\infty)$ and $v_{2z}(t=-\infty)$ are respectively denoted by $v_{1z}$ and $v_{2z}$ for simplicity. This representation will be used in the following without causing any misunderstanding. Now let us briefly talk about the conditions under which reflection occurs. For $p>\rho$, the reflection condition can be readily determined from Eq. (11), $$\begin{alignat}{1} v_{1z}-\sqrt{\frac{2e^2}{4\pi\varepsilon_0\mu p}} < v_{2z} < v_{1z} +\sqrt{\frac{2e^2}{4\pi\varepsilon_0\mu p}}.~~ \tag {16} \end{alignat} $$ For $0 < p < \rho$, $|z|>\rho$ has to be met to ensure the validity of Eq. (11). As a result, the reflection condition in this case is $$\begin{alignat}{1} v_{1z}-\sqrt{\frac{2e^2}{4\pi\varepsilon_0\mu \rho}} < v_{2z} < v_{1z} +\sqrt{\frac{2e^2}{4\pi\varepsilon_0\mu \rho}}.~~ \tag {17} \end{alignat} $$ At this stage, we are in a position to evaluate the electron parallel velocity friction and diffusion coefficients due to e–e collisions with $L_{\rm c}>\rho$. Assume that the electron parallel velocity scattering results from successive binary e–e collisions[18–22] and bear in mind that only e–e collisions with reflections make contributions. Summing over all the collisions electron 1 undergoes per unit time, the expression for the electron parallel velocity friction coefficient $\langle \Delta v_{1z} \rangle_{\rm R}$ reads $$\begin{alignat}{1} \langle \Delta v_{1z}\rangle_{\rm R}\!=\,&\!\!\int_0^{\lambda_{\rm D}}\!\!\!\!\!2\pi pdp\!\!\int_{v_{1z}-\Delta u}^{v_{1z}+\Delta u}\!dv_{2z}\!\int \!\!\!d^2{\boldsymbol v}_{2\perp} f_{\rm e}({\boldsymbol v}_{2\perp}, v_{2z}, t)\\ &\cdot|v_{2z}-v_{1z}|(v_{2z}-v_{1z}),~~ \tag {18} \end{alignat} $$ where ${\boldsymbol v}_{i\perp}$ is the perpendicular component of ${\boldsymbol v}_i$, $f_{\rm e}$ is the electron distribution function, and the integral over $p$ is truncated at $\lambda_{\rm D}$ as a result of the Debye screening. From Eqs. (16) and (17), it follows that $\Delta u=\sqrt{2e^2/(4\pi\varepsilon_0\mu p)}$ for $\bar{\rho} < p < \lambda_{\rm D}$ and $\Delta u=\sqrt{2e^2/(4\pi\varepsilon_0\mu \bar{\rho})}$ for $0 < p < \bar{\rho}$ with $\bar{\rho}$ being the average of $\rho$. For the magnetized plasmas we are concerned with $\bar{\rho}\gg\lambda_{\rm L}$, thus $\Delta u$ is much smaller than the average electron parallel speed and $f_{\rm e}({\boldsymbol v}_{2\perp},v_{2z},t)$ can be Taylor expanded over $v_{2z}$ around $v_{1z}$ as $$\begin{alignat}{1} f_{\rm e}({\boldsymbol v}_{2\perp},v_{2z},t)\simeq\,&f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z},t)+(v_{2z}-v_{1z})\\ &\cdot\frac{\partial f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z},t)}{\partial v_{1z}}.~~ \tag {19} \end{alignat} $$ Substituting the above expansion into Eq. (18), the first term disappears as it is even in $(v_{2z}-v_{1z})$, and the second term yields $$\begin{alignat}{1} \langle \Delta v_{1z}\rangle_{\rm R} =\,&16\pi\Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\Big[\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)+\frac{1}{2}\Big]\\ &\cdot\frac{\partial}{\partial v_{1z}}\int d^2{\boldsymbol v}_{2\perp}f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z},t).~~ \tag {20} \end{alignat} $$ In a similar way, we can obtain the electron parallel velocity diffusion coefficient $\langle\Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}$ $$\begin{align} \langle \Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}=\,&\int_0^{\lambda_{\rm D}}2\pi pdp\int_{v_{1z}-\Delta u}^{v_{1z}+\Delta u}dv_{2z}\int d^2{\boldsymbol v}_{2\perp}\\ &\cdot f_{\rm e}({\boldsymbol v}_{2\perp},v_{2z},t)|v_{2z}-v_{1z}|(v_{2z}-v_{1z})^2\\ =\,& 16\pi\Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\Big[\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)+\frac{1}{2}\Big]\\ &\cdot\int d^2{\boldsymbol v}_{2\perp}f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z},t).~~ \tag {21} \end{align} $$ The terms 1/2 in the square brackets on the right-hand sides of Eqs. (20) and (21) arise from the contribution of collisions with $0 < p < \bar{\rho}$. Within logarithmic accuracy, it will be ignored in the following. If $f_{\rm e}({\boldsymbol v}_2)$ is chosen to be the Maxwellian distribution $$\begin{align} f_{\rm e}({\boldsymbol v}_2)=\frac{n}{(2\pi \bar{v}^2)^{3/2}} \exp\Big(-\frac{v_2^2}{2\bar{v}^2}\Big),~~ \tag {22} \end{align} $$ we will have $$\begin{align} \langle\Delta v_{1z}\rangle_{\rm R}=\,&-8\sqrt{2\pi}n\Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)\\ &\cdot\frac{v_{1z}} {\bar{v}^3}\exp\Big(-\frac{v_{1z}^2}{2\bar{v}^2}\Big),~~ \tag {23} \end{align} $$ $$\begin{align} \langle\Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}=\,&8\sqrt{2\pi}n \Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)\\ &\cdot\frac{1}{\bar{v}}\exp\Big(-\frac{v_{1z}^2}{2\bar{v}^2}\Big).~~ \tag {24} \end{align} $$ It can be seen that $\langle\Delta v_{1z}\rangle_{\rm R}$ and $\langle\Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}$ do not depend on ${\boldsymbol v}_{1\perp}$, as the perpendicular motion of electron 1 is constrained. When $v_{1z}\ll \bar{v}$, we find $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!\langle\Delta v_{1z}\rangle_{\rm R}\,&\approx-8\sqrt{2\pi}n \Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)\frac{v_{1z}}{\bar{v}^3},~~ \tag {25} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!\langle\Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}\,&\approx8\sqrt{2\pi}n \Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)\frac{1}{\bar{v}},~~ \tag {26} \end{alignat} $$ which are, respectively, three times (much larger than) the electron parallel velocity friction and diffusion coefficients arising from the e–e collisions with impact parameters between $\bar{\rho}$ and $\lambda_{\rm D}$ without considering the magnetic field influence on the collisions under the condition $v_{1\perp}\ll \bar{v}$ ($v_{1\perp}\gg \bar{v}$).[23] When $v_{1z}\gg \bar{v}$, $\langle\Delta v_{1z}\rangle_{\rm R}$ and $\langle\Delta v_{1z} \Delta v_{1z}\rangle_{\rm R}$ are exponentially small as the number of the e–e collisions with reflections is exponentially small in this case since Eqs. (16) and (17) indicate that $v_{2z}\approx v_{1z}$ as reflections occur for $L_{\rm c}\gg\lambda_{\rm L}$. Using Eqs. (20) and (21), the electron collision term $\mathcal{C}_{\rm R}$ describing parallel velocity scattering due to e–e collisions with $\bar{\rho} < L_{\rm c} < \lambda_{\rm D}$ can be determined, $$\begin{align} \mathcal{C}_{\rm R}(f_{\rm e}({\boldsymbol v}_1,t))=\,&-\frac{\partial}{\partial v_{1z}}[\langle\Delta v_{1z}\rangle_{\rm R}f_{\rm e}({\boldsymbol v}_1,t)] \\ &+\frac{1}{2}\frac{\partial^2}{\partial v_{1z}\partial v_{1z}}[\langle\Delta v_{1z}\Delta v_{1z}\rangle_{\rm R}f_{\rm e}({\boldsymbol v}_1,t)]\\ =\,&-8\pi\Big(\frac{e^2}{4\pi\varepsilon_0m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big) \frac{\partial}{\partial v_{1z}}\int d^3{\boldsymbol v}_2\\ &\cdot\delta(v_{2z}-v_{1z}) \Big(\frac{\partial}{\partial v_{2z}}-\frac{\partial}{\partial v_{1z}}\Big)\\ &\cdot[f_{\rm e}({\boldsymbol v}_1,t)f_{\rm e}({\boldsymbol v}_2,t)].~~ \tag {27} \end{align} $$ The collision term in the above equation is identical to that obtained by O'Neil[5] except the usual Coulomb logarithm in his formula being replaced by $\ln(\lambda_{\rm D}/\bar{\rho} )$ here. The difference comes from the different ranges of the magnetic field strength and $L_{\rm c}$ considered. It is straightforward to verify that $\mathcal{C}_{\rm R}$ satisfies the conservation of particles, momentum, and energy. The proof procedure closely resembles those given in previous studies and will not be repeated here. Here we concentrate on the final form of $f_{\rm e}$ under the dominance of $\mathcal{C}_{\rm R}$. To this end, the time evolution of the quantity $H$ is traced. $H$ is defined as $$\begin{align} H=\int f_{\rm e}({\boldsymbol v}_1,t)\ln f_{\rm e}({\boldsymbol v}_1,t)d^3{\boldsymbol v}_1.~~ \tag {28} \end{align} $$ Its time rate of change due to $\mathcal{C}_{\rm R}$ is $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\Big(\frac{dH}{dt}\Big)_{\mathcal{C}_{\rm R}}=\int[1+\ln f_{\rm e}({\boldsymbol v}_1,t)]\mathcal{C}_{\rm R}(f_{\rm e}({\boldsymbol v}_1,t))d^3{\boldsymbol v}_1.~~ \tag {29} \end{alignat} $$ Substituting Eq. (27) into the above equation, integrating by parts over $v_{1z}$, interchanging 1 and 2, adding the resulting equivalent expression, and dividing by 2, gives $$\begin{align} \Big(\frac{dH}{dt}\Big)_{\mathcal{C}_{\rm R}}=\,&-4\pi\Big(\frac{e^2}{4\pi\varepsilon_0 m}\Big)^2\ln\Big(\frac{\lambda_{\rm D}}{\bar{\rho}}\Big)\\ &\cdot\int d^3{\boldsymbol v}_1\int d^3{\boldsymbol v}_2f_{\rm e}({\boldsymbol v}_1,t)f_{\rm e}({\boldsymbol v}_2,t)\\ &\cdot\Big[\frac{\partial \ln f_{\rm e}({\boldsymbol v}_1,t)}{\partial v_{1z}}-\frac{\partial \ln f_{\rm e}({\boldsymbol v}_2,t)}{\partial v_{2z}}\Big]^2\\ &\cdot\delta(v_{2z}-v_{1z})\leqq 0.~~ \tag {30} \end{align} $$ The equality will hold if and only if $$\begin{alignat}{1} \frac{\partial\ln f_{\rm e}({\boldsymbol v}_{1\perp},v_{1z})}{\partial v_{1z}}-\frac{\partial\ln f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z})}{\partial v_{1z}}=0.~~ \tag {31} \end{alignat} $$ Integrating over $v_{1z}$ gives $$\begin{align} \frac{f_{\rm e}({\boldsymbol v}_{1\perp},v_{1z})}{f_{\rm e}({\boldsymbol v}_{2\perp},v_{1z})}=C({\boldsymbol v}_{1\perp},{\boldsymbol v}_{2\perp}).~~ \tag {32} \end{align} $$ The solution of the above equation is $$\begin{align} f_{\rm e}({\boldsymbol v}_{\perp},v_{z})=n F_{\rm e}({\boldsymbol v}_{\perp})G_{\rm e}(v_{z}),~~ \tag {33} \end{align} $$ where $$\begin{align} F_{\rm e}({\boldsymbol v}_{\perp})=\,&\frac{1}{n}\int_{-\infty}^\infty f_{\rm e}({\boldsymbol v}_{\perp},v_{z})dv_z,~~ \tag {34} \end{align} $$ $$\begin{align} G_{\rm e}(v_{z})=\,&\frac{1}{n}\int f_{\rm e}({\boldsymbol v}_{\perp},v_{z})d^2{\boldsymbol v}_\perp.~~ \tag {35} \end{align} $$ It can be readily verified that $f_{\rm e}$ of the form in Eq. (33) satisfies $\mathcal{C}_{\rm R}(f_{\rm e})=0$. It follows that $\mathcal{C}_{\rm R}$ does not necessarily make the system relax to the Maxwellian distribution, which is a little surprising. Recall that guiding center approximation is utilized in deriving $C_{\rm R}$, which prohibits the energy transfer between the parallel and perpendicular degrees of freedom. When the velocity change in the collision is evaluated more accurately, keeping higher order terms in $\rho/L_{\rm c}$, the e–e collisions with reflections are also believed to relax the system to the Maxwellian distribution eventually. This issue is beyond the scope of the present work. From Eq. (27), it follows that $$\begin{align} \int_{-\infty}^\infty \mathcal{C}_{\rm R}(f_{\rm e}({\boldsymbol v},t))dv_z=\,&0,~~ \tag {36} \end{align} $$ $$\begin{align} \int \mathcal{C}_{\rm R}(f_{\rm e}({\boldsymbol v},t))d^2{\boldsymbol v}_\perp=\,&0,~~ \tag {37} \end{align} $$ thus in the scenario where the temporal evolution of $f_{\rm e}$ is totally governed by $\mathcal{C}_{\rm R}(f_{\rm e})$, $F_{\rm e}({\boldsymbol v}_{\perp})$ and $G_{\rm e}(v_{z})$ can be determined from the initial distribution $f_{\rm e}({\boldsymbol v},0)$ as $$\begin{align} F_{\rm e}({\boldsymbol v}_{\perp})=\,&\frac{1}{n}\int_{-\infty}^\infty f_{\rm e}({\boldsymbol v},0)dv_z,~~ \tag {38} \end{align} $$ $$\begin{align} G_{\rm e}(v_{z})=\,&\frac{1}{n}\int f_{\rm e}({\boldsymbol v},0)d^2{\boldsymbol v}_\perp.~~ \tag {39} \end{align} $$ Accordingly, whether the system eventually reaches the Maxwellian state depends entirely on the initial distribution under this condition. In summary, the electron collision term $\mathcal{C}_{\rm R}$ describing parallel velocity scattering due to reflections in e–e collisions with $\bar{\rho} < L_{\rm c} < \lambda_{\rm D}$ has been derived based on the Fokker–Planck approach. Since the electron perpendicular motion is constrained for these collisions, the evaluated electron parallel velocity friction coefficient $\langle\Delta v_z\rangle_{\rm R}$ and diffusion coefficient $\langle \Delta v_z \Delta v_z\rangle_{\rm R}$ are shown not to depend on the electron perpendicular velocity. The time evolution of the $H$ quantity due to $\mathcal{C}_{\rm R}$ is studied. It is found that $\mathcal{C}_{\rm R}$ does not necessarily relax $f_{\rm e}$ to the Maxwellian distribution but to the form $f_{\rm e}({\boldsymbol v}_\perp,v_z)=F_{\rm e}({\boldsymbol v}_\perp) G_{\rm e}(v_z)$. 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