From Blelly and Schunk (1993), the 13-moment equation set gives:

$\frac{\delta T^{\perp}_i}{\delta t} = \sum_t \frac{\nu_{it}}{m_i+m_t}\left[2m_i(T^{\perp}_t-T^{\perp}_i) + \frac{2}{5}m_t(T^{\parallel}_i-T^{\perp}_i) + \frac{2}{5}m_i(T^{\parallel}_t-T^{\perp}_t)\right] + \frac{2}{5}\nu_{ii}(T^{\parallel}_i-T^{\perp}_i)$

$\begin{eqnarray} \frac{\delta T^{\perp}_i}{\delta t} &=& \sum_n \frac{\nu_{in}}{m_i+m_n}\left[2m_i(T^{\perp}_n-T^{\perp}_i) + \frac{2}{5}m_n(T^{\parallel}_i-T^{\perp}_i) + \frac{2}{5}m_i(T^{\parallel}_n-T^{\perp}_n)\right] \\ &+& \sum_{j \ne i} \frac{\nu_{ij}}{m_i+m_j}\left[2m_i(T^{\perp}_j-T^{\perp}_i) + \frac{2}{5}m_j(T^{\parallel}_i-T^{\perp}_i) + \frac{2}{5}m_i(T^{\parallel}_j-T^{\perp}_j)\right] \\ &+& \frac{\nu_{ie}}{m_i+m_e}\left[2m_i(T^{\perp}_e-T^{\perp}_i) + \frac{2}{5}m_e(T^{\parallel}_i-T^{\perp}_i) + \frac{2}{5}m_i(T^{\parallel}_e-T^{\perp}_e)\right] \\ &+& \frac{2}{5}\nu_{ii}(T^{\parallel}_i-T^{\perp}_i) \end{eqnarray}$

Contribution $T^{\perp}_i \rightarrow T^{\perp}_i$ :

$\begin{eqnarray} -2\sum_n \nu_{in}\frac{m_i}{m_i+m_n}T^{\perp}_i - \frac{2}{5}\sum_n \nu_{in}\frac{m_n}{m_i+m_n}T^{\perp}_i &=& -\sum_n \nu_{in}\frac{2m_i-\frac{2}{5}m_n}{m_i+m_n}T^{\perp}_i\\ -2\sum_{j \ne i}\nu_{ij}\frac{m_i}{m_i+m_j}T^{\perp}_i - \frac{2}{5}\sum_{j \ne i} \nu_{ij}\frac{m_j}{m_i+m_j}T^{\perp}_i &=& -\sum_{j \ne i} \nu_{ij}\frac{2m_i-\frac{2}{5}m_j}{m_i+m_j}T^{\perp}_i\\ -2 \nu_{ie}\frac{m_i}{m_i+m_e}T^{\perp}_i - \frac{2}{5} \nu_{ie}\frac{m_e}{m_i+m_e}T^{\perp}_i &=& - \nu_{ie}\frac{2m_i-\frac{2}{5}m_e}{m_i+m_e}T^{\perp}_i\\ - \frac{2}{5}\nu_{ii}T^{\perp}_i && \end{eqnarray}$

$\text{contrib} \left( T^{\parallel}_i \rightarrow T^{\parallel}_i \right) + \text{contrib} \left( T^{\parallel}_i \rightarrow T^{\perp}_i \right)$

Contribution $T^{\parallel}_i \rightarrow T^{\perp}_i$ :

$\underbrace{ \frac{2}{5}\sum_n \nu_{in}\frac{m_n}{m_i+m_n}T^{\parallel}_i + \frac{2}{5}\sum_{j \ne i} \nu_{ij}\frac{m_j}{m_i+m_j}T^{\parallel}_i + \frac{2}{5} \nu_{ie}\frac{m_e}{m_i+m_e}T^{\parallel}_i + \frac{2}{5} \nu_{ii}T^{\parallel}_i }_{\frac{1}{2}\text{contrib} \left( T^{\perp}_i \rightarrow T^{\parallel}_i \right)}$

$\begin{eqnarray} \sum_n (2-\frac{2}{5})\underbrace{\nu_{in}\frac{m_i}{m_i+m_n}}_{\text{coefin}}T^{\perp}_n & \text{ Contribution } T^{\perp}_n \rightarrow T^{\perp}_i\\ \sum_{j \ne i} (2-\frac{2}{5})\underbrace{\nu_{ij}\frac{m_i}{m_i+m_j}}_{\text{coefij}}T^{\perp}_j & \text{ Contribution } T^{\perp}_j \rightarrow T^{\perp}_i\\ (2-\frac{2}{5})\underbrace{\nu_{ie}\frac{m_i}{m_i+m_e}}_{\text{coefie}}T^{\perp}_e & \text{ Contribution } T^{\perp}_e \rightarrow T^{\perp}_i\\ \sum_n \frac{2}{5}\underbrace{\nu_{in}\frac{m_i}{m_i+m_n}}_{\text{coefin}}T^{\parallel}_n & \text{ Contribution } T^{\parallel}_n \rightarrow T^{\perp}_i\\ \sum_{j \ne i} \frac{2}{5}\underbrace{\nu_{ij}\frac{m_i}{m_i+m_j}}_{\text{coefij}}T^{\parallel}_j & \text{ Contribution } T^{\parallel}_j \rightarrow T^{\perp}_i\\ \frac{2}{5}\underbrace{\nu_{ie}\frac{m_i}{m_i+m_e}}_{\text{coefie}}T^{\parallel}_e & \text{ Contribution } T^{\parallel}_e \rightarrow T^{\perp}_i \end{eqnarray}$