For ions $i$ :

$\begin{eqnarray*} \frac{\delta \gamma_i^{\parallel}}{\delta t} &=& t_i \sum_n \nu_{in}A^{m0p}_{in}(\frac{C_n}{C_i} u_n - u_i)\frac{1}{t_i^{\parallel\frac{3}{2}}}\\ &+& t_i \sum_{j \ne i} \nu_{ij}A^{c0p}_{ij}(\frac{C_j}{C_i} u_j - u_i)\frac{1}{t_i^{\parallel\frac{3}{2}}}\\ &+& t_i \nu_{ie}A^{c0p}_{ie}(\frac{1 }{C_i} U_e - u_i)\frac{1}{t_i^{\parallel\frac{3}{2}}}\\ &+& \sum_n \nu_{in}\left( D^{m1pt}_{in} \frac{t_i^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_i^{\perp }+ D^{m4pt}_{in}\frac{C_n}{C_i} \frac{t_n^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_n^{\perp }+ D^{m1pp}_{in} \gamma_i^{\parallel }+ D^{m4pp}_{in}\frac{C_n}{C_i} \frac{t_n^{\parallel \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_n^{\parallel }\right)\\ &+& \sum_{j \ne i} \nu_{ij}\left( D^{c1pt}_{ij} \frac{t_i^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_i^{\perp }+ D^{c4pt}_{ij}\frac{C_j}{C_i} \frac{t_j^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_j^{\perp }+ D^{c1pp}_{ij} \gamma_i^{\parallel }+ D^{c4pp}_{ij}\frac{C_j}{C_i} \frac{t_j^{\parallel \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_j^{\parallel }\right)\\ &+& \nu_{ii}\left( D^{c1pt}_{ii} \frac{t_i^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_i^{\perp }+ D^{c4pt}_{ii} \frac{t_i^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_i^{\perp }+ D^{c1pp}_{ii} \gamma_i^{\parallel }+ D^{c4pp}_{ii} \gamma_i^{\parallel }\right)\\ &+& \nu_{ie}\left( D^{c1pt}_{ie} \frac{t_i^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_i^{\perp }+ D^{c4pt}_{ie}\frac{C_e}{C_i} \frac{t_e^{\perp \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_e^{\perp }+ D^{c1pp}_{ie} \gamma_i^{\parallel }+ D^{c4pp}_{ie}\frac{C_e}{C_i} \frac{t_e^{\parallel \frac{3}{2}}}{t_i^{\parallel \frac{3}{2}}} \gamma_e^{\parallel }\right)\\ &-& \left.3t_i^{\parallel}\frac{\delta u_i}{\delta t}\frac{1}{t_i^{\parallel\frac{3}{2}}}\right]\\ \end{eqnarray*}$

$\begin{eqnarray*} t_i\sum_n \nu_{in}A^{m0p}_{in}u_n\frac{C_n}{C_i} \frac{1}{t_i^{\parallel\frac{3}{2}}} - 3t_i^{\parallel}\sum_n \nu_{in}u_n\frac{C_n}{C_i} \frac{1}{t_i^{\parallel\frac{3}{2}}} & \text{ Contribution } u_n \rightarrow \gamma_i^{\parallel}\\ t_i\sum_{j \ne i} \nu_{ij}A^{c0p}_{ij}u_j\frac{C_j}{C_i} \frac{1}{t_i^{\parallel\frac{3}{2}}} - 3t_i^{\parallel}\sum_{j \ne i} \nu_{ij}u_j\frac{C_j}{C_i} \frac{1}{t_i^{\parallel\frac{3}{2}}} & \text{ Contribution } u_j \rightarrow \gamma_i^{\parallel}\\ t_i \nu_{ie}A^{c0p}_{ie}\left(compo_i u_i + \sum_{j \ne i}compo_j u_j\frac{C_j}{C_i}\right) \frac{1}{t_i^{\parallel\frac{3}{2}}} - 3t_i^{\parallel} \nu_{ie}\left(compo_i u_i + \sum_{j \ne i}compo_j u_j\frac{C_j}{C_i}\right) \frac{1}{t_i^{\parallel\frac{3}{2}}} & \text{ Contribution } u_e \rightarrow \gamma_i^{\parallel}\\ \sum_n \nu_{in}D^{m4pp}_{in}\frac{C_n}{C_i}\frac{t_n^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_n^{\parallel} + \frac{3}{2}t_i^{\parallel}\sum_n \underbrace{D^{mo}_{ni}\nu_{in}\frac{1}{t_{in}}}_{\text{coefin}}\frac{C_n}{C_i}\frac{t_n^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_n^{\parallel} & \text{ Contribution } \gamma_n^{\parallel} \rightarrow \gamma_i^{\parallel}\\ \sum_{j \ne i} \nu_{ij}D^{c4pp}_{ij}\frac{C_j}{C_i}\frac{t_j^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_j^{\parallel} + \frac{3}{2}t_i^{\parallel}\sum_{j \ne i} \underbrace{D^{co}_{ji}\nu_{ij}\frac{1}{t_{ij}}}_{\text{coefij}}\frac{C_j}{C_i}\frac{t_j^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_j^{\parallel} & \text{ Contribution } \gamma_j^{\parallel} \rightarrow \gamma_i^{\parallel}\\ \nu_{ie}D^{c4pp}_{ie}\frac{C_e}{C_i}\frac{t_e^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_e^{\parallel} + \frac{3}{2}t_i^{\parallel} \underbrace{D^{co}_{ei}\nu_{ie}\frac{1}{t_{ie}}}_{\text{coefie}}\frac{C_e}{C_i}\frac{t_e^{\parallel\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_e^{\parallel} & \text{ Contribution } \gamma_e^{\parallel} \rightarrow \gamma_i^{\parallel}\\ \sum_n \nu_{in}D^{m4pt}_{in}\frac{C_n}{C_i}\frac{t_n^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_n^{\perp} + 3t_i^{\parallel}\sum_n \underbrace{D^{mo}_{ni}\nu_{in}\frac{1}{t_{in}}}_{\text{coefin}}\frac{C_n}{C_i}\frac{t_n^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_n^{\perp} & \text{ Contribution } \gamma_n^{\perp} \rightarrow \gamma_i^{\parallel}\\ \sum_{j \ne i} \nu_{ij}D^{c4pt}_{ij}\frac{C_j}{C_i}\frac{t_j^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_j^{\perp} + 3t_i^{\parallel}\sum_{j \ne i} \underbrace{D^{co}_{ji}\nu_{ij}\frac{1}{t_{ij}}}_{\text{coefij}}\frac{C_j}{C_i}\frac{t_j^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_j^{\perp} & \text{ Contribution } \gamma_j^{\perp} \rightarrow \gamma_i^{\parallel}\\ \nu_{ie}D^{c4pt}_{ie}\frac{C_n}{C_i}\frac{t_e^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_e^{\perp} + 3t_i^{\parallel} \underbrace{D^{co}_{ei}\nu_{ie}\frac{1}{t_{ie}}}_{\text{coefie}}\frac{C_e}{C_i}\frac{t_e^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_e^{\perp} & \text{ Contribution } \gamma_e^{\perp} \rightarrow \gamma_i^{\parallel}\\ \end{eqnarray*}$

Contribution $u_i \rightarrow \gamma_i^{\parallel}$ :

$-t_i\left( \underbrace{\sum_n \nu_{in}A^{m0p}_{in} }_{\text{Dm0pnui}}+ \underbrace{\sum_{j \ne i} \nu_{ij}A^{c0p}_{ij}+\nu_{ie}A^{c0p}_{ie} }_{\text{Dc0pnui}} \right)\frac{1}{t_i^{\parallel\frac{3}{2}}} +3t_i^{\parallel}\left(\underbrace{\sum_n \nu_{in} + \sum_{j \ne i} \nu_{ij} + \nu_{ie}}_{\text{nuin} + \text{nuij}}\right)\frac{1}{t_i^{\parallel\frac{3}{2}}}$

Contribution $\gamma_i^{\parallel} \rightarrow \gamma_i^{\parallel}$ :

$\left( \sum_n \nu_{in}D^{m1pp}_{in} + \underbrace{ \sum_{j \ne i} \nu_{ij}D^{c1pp}_{ij} + \nu_{ie}D^{c1pp}_{ie} + \nu_{ii}D^{c1pp}_{ii} + \nu_{ii}D^{c4pp}_{ii}}_{\text{Dc1ppnui}} \right)\gamma_i^{\parallel} -\frac{3}{2}t_i^{\parallel}\left(\text{Dmnui} + \text{Dcnui}\right)\gamma_i^{\parallel}$

Contribution $\gamma_i^{\perp} \rightarrow \gamma_i^{\parallel}$ :

$\left( \sum_n \nu_{in}D^{m1pt}_{in} + \underbrace{ \sum_{j \ne i} \nu_{ij}D^{c1pt}_{ij} + \nu_{ie}D^{c1pt}_{ie} + \nu_{ii}D^{c1pt}_{ii} + \nu_{ii}D^{c4pt}_{ii}}_{\text{Dc1ptnui}} \right)\frac{t_i^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_i^{\perp} -3t_i^{\parallel}\left(\text{Dmnui} + \text{Dcnui}\right)\frac{t_i^{\perp\frac{3}{2}}}{t_i^{\parallel\frac{3}{2}}}\gamma_i^{\perp}$