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

$\frac{\delta T^{\perp}_n}{\delta t} = \sum_t \frac{\nu_{nt}}{m_n+m_t}\left[2m_n(T^{\perp}_t-T^{\perp}_n) + \frac{2}{5}m_t(T^{\parallel}_n-T^{\perp}_n) + \frac{2}{5}m_n(T^{\parallel}_t-T^{\perp}_t)\right] + \frac{2}{5}\nu_{nn}(T^{\parallel}_n-T^{\perp}_n)$

$\begin{eqnarray} \frac{\delta T^{\perp}_n}{\delta t} &=& \sum_{m \ne n} \frac{\nu_{nm}}{m_n+m_m}\left[2m_n(T^{\perp}_m-T^{\perp}_n) + \frac{2}{5}m_m(T^{\parallel}_n-T^{\perp}_n) + \frac{2}{5}m_n(T^{\parallel}_m-T^{\perp}_m)\right] \\ &+& \sum_i \frac{\nu_{ni}}{m_n+m_i}\left[2m_n(T^{\perp}_i-T^{\perp}_n) + \frac{2}{5}m_i(T^{\parallel}_n-T^{\perp}_n) + \frac{2}{5}m_n(T^{\parallel}_i-T^{\perp}_i)\right] \\ &+& \frac{\nu_{ne}}{m_n+m_e}\left[2m_n(T^{\perp}_e-T^{\perp}_n) + \frac{2}{5}m_e(T^{\parallel}_n-T^{\perp}_n) + \frac{2}{5}m_n(T^{\parallel}_e-T^{\perp}_e)\right] \\ &+& \frac{2}{5}\nu_{nn}(T^{\parallel}_n-T^{\perp}_n) \end{eqnarray}$

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

$\begin{eqnarray} -2\sum_{m \ne n}\nu_{nm}\frac{m_n}{m_n+m_m}T^{\perp}_n - \frac{2}{5}\sum_{m \ne n} \nu_{nm}\frac{m_m}{m_n+m_m}T^{\perp}_n &=& -\sum_{m \ne n} \nu_{nm}\frac{2m_n-\frac{2}{5}m_m}{m_n+m_m}T^{\perp}_n\\ -2\sum_i \nu_{ni}\frac{m_n}{m_n+m_i}T^{\perp}_n - \frac{2}{5}\sum_i \nu_{ni}\frac{m_i}{m_n+m_i}T^{\perp}_n &=& -\sum_i \nu_{ni}\frac{2m_n-\frac{2}{5}m_i}{m_n+m_i}T^{\perp}_n\\ -2 \nu_{ne}\frac{m_n}{m_n+m_e}T^{\perp}_n - \frac{2}{5} \nu_{ne}\frac{m_e}{m_n+m_e}T^{\perp}_n &=& - \nu_{ne}\frac{2m_n-\frac{2}{5}m_e}{m_n+m_e}T^{\perp}_n\\ - \frac{2}{5}\nu_{nn}T^{\perp}_n && \end{eqnarray}$

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

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

$\underbrace{ \frac{2}{5}\sum_{m \ne n} \nu_{nm}\frac{m_m}{m_n+m_m}T^{\parallel}_n + \frac{2}{5}\sum_i \nu_{ni}\frac{m_i}{m_n+m_i}T^{\parallel}_n + \frac{2}{5} \nu_{ne}\frac{m_e}{m_n+m_e}T^{\parallel}_n + \frac{2}{5} \nu_{nn}T^{\parallel}_n }_{\frac{1}{2}\text{contrib} \left( T^{\perp}_n \rightarrow T^{\parallel}_n \right)}$

$\begin{eqnarray} \sum_{m \ne n} (2-\frac{2}{5})\underbrace{\nu_{nm}\frac{m_n}{m_n+m_m}}_{\text{coefnm}}T^{\perp}_m & \text{ Contribution } T^{\perp}_m \rightarrow T^{\perp}_n\\ \sum_i (2-\frac{2}{5})\underbrace{\nu_{ni}\frac{m_n}{m_n+m_i}}_{\text{coefni}}T^{\perp}_i & \text{ Contribution } T^{\perp}_i \rightarrow T^{\perp}_n\\ (2-\frac{2}{5})\underbrace{\nu_{ne}\frac{m_n}{m_n+m_e}}_{\text{coefne}}T^{\perp}_e & \text{ Contribution } T^{\perp}_e \rightarrow T^{\perp}_n\\ \sum_{m \ne n} \frac{2}{5}\underbrace{\nu_{nm}\frac{m_n}{m_n+m_m}}_{\text{coefnm}}T^{\parallel}_m & \text{ Contribution } T^{\parallel}_m \rightarrow T^{\perp}_n\\ \sum_i \frac{2}{5}\underbrace{\nu_{ni}\frac{m_n}{m_n+m_i}}_{\text{coefni}}T^{\parallel}_i & \text{ Contribution } T^{\parallel}_i \rightarrow T^{\perp}_n\\ \frac{2}{5}\underbrace{\nu_{ne}\frac{m_n}{m_n+m_e}}_{\text{coefne}}T^{\parallel}_e & \text{ Contribution } T^{\parallel}_e \rightarrow T^{\perp}_n \end{eqnarray}$