For ions $i$ :

$\begin{eqnarray*} \frac{\delta u_i}{\delta t} &=& \sum_n \nu_{in}(u_n\frac{C_n}{C_i}-u_i)\\ &+& \sum_{j \ne i} \nu_{ij}(u_j\frac{C_j}{C_i}-u_i)\\ &+& \nu_{ie}(U_e\frac{1}{C_i }-u_i)\\ &+& \sum_n \frac{\nu_{in}z_{in}}{t_{in}(m_i+m_n)} \left( \frac{1}{2} m_n t_i^{\parallel \frac{3}{2}}\gamma_i^{\parallel } + m_n t_i^{\perp \frac{3}{2}}\gamma_i^{\perp } -\frac{1}{2} m_i \frac{C_n}{C_i} t_n^{\parallel \frac{3}{2}}\gamma_n^{\parallel } - m_i \frac{C_n}{C_i} t_n^{\perp \frac{3}{2}}\gamma_n^{\perp } \right)\\ &+& \sum_{j \ne i} \frac{\nu_{ij}z_{ij}}{t_{ij}(m_i+m_j)} \left( \frac{1}{2} m_j t_i^{\parallel \frac{3}{2}}\gamma_i^{\parallel } + m_j t_i^{\perp \frac{3}{2}}\gamma_i^{\perp } -\frac{1}{2} m_i \frac{C_j}{C_i} t_j^{\parallel \frac{3}{2}}\gamma_j^{\parallel } - m_i \frac{C_j}{C_i} t_j^{\perp \frac{3}{2}}\gamma_j^{\perp } \right)\\ &+& \frac{\nu_{ie}z_{ie}}{t_{ie}(m_i+m_e)} \left( \frac{1}{2} m_e t_i^{\parallel \frac{3}{2}}\gamma_i^{\parallel } + m_e t_i^{\perp \frac{3}{2}}\gamma_i^{\perp } -\frac{1}{2} m_i \frac{C_e}{C_i} t_e^{\parallel \frac{3}{2}}\gamma_e^{\parallel } - m_i \frac{C_e}{C_i} t_e^{\perp \frac{3}{2}}\gamma_e^{\perp } \right)\\ \end{eqnarray*}$

$\begin{eqnarray*} \underbrace{-\sum_n \nu_{in}}_{\text{-nuin}} \underbrace{-\sum_{j \ne i} \nu_{ij} - \nu_{ie}}_{\text{-nuij}} & \text{ Contribution } u_i \rightarrow u_i\\ \nu_{in}u_n\frac{C_n}{C_i} & \text{ Contribution } u_n \rightarrow u_i\\ \nu_{ij}u_j\frac{C_j}{C_i} & \text{ Contribution } u_j \rightarrow u_i\\ \nu_{ie}U_e\frac{1 }{C_i} = \nu_{ie}\left(compo_i u_i + \sum_{j \ne i} compo_j u_j\frac{C_j}{C_i}\right) & \text{ Contribution } u_e \rightarrow u_i\\ -\frac{1}{2}\sum_n z_{in}\frac{m_i}{m_i+m_n}\nu_{in}\frac{1}{t_{in}}\frac{C_n}{C_i}t_n^{\parallel\frac{3}{2}}\gamma_n^{\parallel} = -\frac{1}{2}\sum_n \underbrace{D^{mo}_{ni}\nu_{in}\frac{1}{t_{in}}}_{\text{coefin}}\frac{C_n}{C_i}t_n^{\parallel\frac{3}{2}}\gamma_n^{\parallel} &\text{ Contribution }\gamma_n^{\parallel}\rightarrow u_i\\ -\frac{1}{2}\sum_{j \ne i} z_{ij}\frac{m_i}{m_i+m_j}\nu_{ij}\frac{1}{t_{ij}}\frac{C_j}{C_i}t_j^{\parallel\frac{3}{2}}\gamma_j^{\parallel} = -\frac{1}{2}\sum_{j \ne i} \underbrace{D^{co}_{ji}\nu_{ij}\frac{1}{t_{ij}}}_{\text{coefij}}\frac{C_j}{C_i}t_j^{\parallel\frac{3}{2}}\gamma_j^{\parallel} &\text{ Contribution }\gamma_j^{\parallel}\rightarrow u_i\\ -\frac{1}{2} z_{ie}\frac{m_i}{m_i+m_e}\nu_{ie}\frac{1}{t_{ie}}\frac{C_e}{C_i}t_e^{\parallel\frac{3}{2}}\gamma_e^{\parallel} = -\frac{1}{2} \underbrace{D^{co}_{ei}\nu_{ie}\frac{1}{t_{ie}}}_{\text{coefie}}\frac{C_e}{C_i}t_e^{\parallel\frac{3}{2}}\gamma_e^{\parallel} &\text{ Contribution }\gamma_e^{\parallel}\rightarrow u_i\\ -\sum_n z_{in}\frac{m_i}{m_i+m_n}\nu_{in}\frac{1}{t_{in}}\frac{C_n}{C_i}t_n^{\perp\frac{3}{2}}\gamma_n^{\perp} = -\sum_n \underbrace{D^{mo}_{ni}\nu_{in}\frac{1}{t_{in}}}_{\text{coefin}}\frac{C_n}{C_i}t_n^{\perp\frac{3}{2}}\gamma_n^{\perp} &\text{ Contribution }\gamma_n^{\perp}\rightarrow u_i\\ -\sum_{j \ne i} z_{ij}\frac{m_i}{m_i+m_j}\nu_{ij}\frac{1}{t_{ij}}\frac{C_j}{C_i}t_j^{\perp\frac{3}{2}}\gamma_j^{\perp} = -\sum_{j \ne i} \underbrace{D^{co}_{ji}\nu_{ij}\frac{1}{t_{ij}}}_{\text{coefij}}\frac{C_j}{C_i}t_j^{\perp\frac{3}{2}}\gamma_j^{\perp} &\text{ Contribution }\gamma_j^{\perp}\rightarrow u_i\\ - z_{ie}\frac{m_i}{m_i+m_e}\nu_{ie}\frac{1}{t_{ie}}\frac{C_e}{C_i}t_e^{\perp\frac{3}{2}}\gamma_e^{\perp} = - \underbrace{D^{co}_{ei}\nu_{ie}\frac{1}{t_{ie}}}_{\text{coefie}}\frac{C_e}{C_i}t_e^{\perp\frac{3}{2}}\gamma_e^{\perp} &\text{ Contribution }\gamma_e^{\perp}\rightarrow u_i\\ \end{eqnarray*}$

Contribution $\gamma^{\parallel}_i \rightarrow u_i$ :

$\left.\begin{aligned} \frac{1}{2}\sum_n z_{in}\frac{m_n}{m_i+m_n}\nu_{in}\frac{1}{t_{in}}t_i^{\parallel\frac{3}{2}} &=& \frac{1}{2}\left(\sum_n D^{mo}_{in}\nu_{in}\frac{1}{t_{in}}\right)t_i^{\parallel\frac{3}{2}} \end{aligned}\right\} \frac{1}{2}\text{Dmnui.}t_i^{\parallel\frac{3}{2}}$

$\left.\begin{aligned} \frac{1}{2}\sum_{j \ne i} z_{ij}\frac{m_j}{m_i+m_j}\nu_{ij}\frac{1}{t_{ij}}t_i^{\parallel\frac{3}{2}} &=& \frac{1}{2}\left(\sum_{j \ne i} D^{co}_{ij}\nu_{ij}\frac{1}{t_{ij}}\right)t_i^{\parallel\frac{3}{2}}\\ \frac{1}{2} z_{ie}\frac{m_e}{m_i+m_e}\nu_{ie}\frac{1}{t_{ie}}t_i^{\parallel\frac{3}{2}} &=& \frac{1}{2}\left( D^{co}_{ie}\nu_{ie}\frac{1}{t_{ie}}\right)t_i^{\parallel\frac{3}{2}}\\ \end{aligned}\right\} \frac{1}{2}\text{Dcnui.}t_i^{\parallel\frac{3}{2}}$

Contribution $\gamma^{\perp}_i \rightarrow u_i$ :

$\left.\begin{aligned} \sum_n z_{in}\frac{m_n}{m_i+m_n}\nu_{in}\frac{1}{t_{in}}t_i^{\perp\frac{3}{2}} &=& \left(\sum_n D^{mo}_{in}\nu_{in}\frac{1}{t_{in}}\right)t_i^{\perp\frac{3}{2}}\\ \end{aligned}\right\} \text{Dmnui.}t_i^{\perp\frac{3}{2}}$

$\left.\begin{aligned} \sum_{j \ne i} z_{ij}\frac{m_j}{m_i+m_j}\nu_{ij}\frac{1}{t_{ij}}t_i^{\perp\frac{3}{2}} &=& \left(\sum_{j \ne i} D^{co}_{ij}\nu_{ij}\frac{1}{t_{ij}}\right)t_i^{\perp\frac{3}{2}}\\ z_{ie}\frac{m_e}{m_i+m_e}\nu_{ie}\frac{1}{t_{ie}}t_i^{\perp\frac{3}{2}} &=& \left( D^{co}_{ie}\nu_{ie}\frac{1}{t_{ie}}\right)t_i^{\perp\frac{3}{2}}\\ \end{aligned}\right\} \text{Dcnui.}t_i^{\perp\frac{3}{2}}$