The anisotropic conduction equation looks like

\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \chi_\parallel - \chi_\perp \right ) \left ( \hat{b} \cdot \nabla T \right ) + n \chi_\perp \nabla T \right ]

however, the conduction coefficients have a Temperature dependence.

\chi_\parallel = \kappa_\parallel T^{5/2}

\chi_\perp = \kappa_\perp \frac{n}{B^2 T^{1/2}}

Let's first just consider the \chi_\parallel term.

\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \kappa_\parallel T^\lambda \left ( \hat{b} \cdot \nabla T \right )\right ]

We need a way to write this implicitly but we need it to also be linear in T_{*}

\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \kappa_\parallel T_{*}^\lambda \left ( \hat{b} \cdot \nabla T_{*} \right )\right ]

We could Taylor expand T^\lambda_{*}=T^\lambda + \lambda T^{\lambda-1} \left ( T_{*}-T \right ) = \left ( 1-\lambda \right ) T^\lambda + \lambda T^{\lambda-1} T_{*}

but then we would still have a non-linear term like \lambda T^{\lambda-1} T_{*} \nabla T_{*}

We could write this as \lambda T^{\lambda-1} 1/2 \nabla T_{*}^2 and Taylor expand again to get \lambda T^{\lambda-1} 1/2 \nabla \left ( - T + 2 TT_{*} \right )

but we've now done a Taylor expansion on a Taylor expansion…

Alternatively, we can rewrite the diffusion equation

\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla T^{\lambda+1} \right )\right ]

and then perform a single Taylor expansion

\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla \left (-\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right ) \right ) \right ]

Switching to Einstein notation, we have

\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )

\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \left [ \left ( \partial_i n b_i b_j \right ) \left ( -\lambda \partial j T^{\lambda+1} + \left ( \lambda + 1 \right ) \partial j T^{\lambda}T_{*} \right ) + n b_i b_j \left ( -\lambda \partial_i \partial j T^{\lambda+1} + \left ( \lambda + 1 \right ) \partial_i \partial j T^{\lambda}T_{*} \right ) \right ]