Cartesian Coordinates

The Fokker-Planck form of the focused transport equation is

\[\begin{split}\begin{aligned} \frac{\partial F}{\partial t} = & \nabla\cdot\left[\nabla\cdot(\boldsymbol{\kappa}_\perp\nabla F)\right] - \nabla\cdot\left[(v\mu\boldsymbol{b} + \boldsymbol{V} + \boldsymbol{V}_d + \nabla\cdot\boldsymbol{\kappa}_\perp)F\right] \nonumber\\ & + \frac{\partial}{\partial\mu^2}\left(D_{\mu\mu}F\right) - \frac{\partial}{\partial\mu}\left[\left(\frac{d\mu}{dt}+\frac{\partial D_{\mu\mu}}{\partial\mu}\right)F\right] - \frac{\partial}{\partial p}\left(\frac{dp}{dt}F\right) \end{aligned}\end{split}\]

where \(F=fp^2\). The corresponding SDEs are

\[\begin{split}\begin{aligned} d\boldsymbol{X} & = (v\mu\boldsymbol{b} + \boldsymbol{V} + \boldsymbol{V}_d + \nabla\cdot\boldsymbol{\kappa}_\perp)dt + \sum_\sigma\boldsymbol{\alpha}_\sigma dW_\sigma(s) \\ dp & = \left(\frac{dp}{dt}\right)dt \\ d\mu & = \left(\frac{d\mu}{dt}+\frac{\partial D_{\mu\mu}}{\partial\mu}\right)dt + \sqrt{2D_{\mu\mu}}dW_\mu(t) \end{aligned}\end{split}\]

where \(\sum_\sigma\alpha_\sigma^\mu\alpha_\sigma^\nu = 2\kappa_\perp^{\mu\nu}\). We can use the results from the 3D model with whole spatial diffusion tensor \(\boldsymbol{\kappa}\) but set \(\kappa_\parallel=0\). The matrix

\[\begin{split}P = \begin{pmatrix} 0 & -b_xb_z\sqrt{2\kappa_\perp}/\sqrt{b_x^2+b_y^2} & -b_y\sqrt{2\kappa_\perp}/\sqrt{b_x^2+b_y^2}\\ 0 & -b_yb_z\sqrt{2\kappa_\perp}/\sqrt{b_x^2+b_y^2} & b_x\sqrt{2\kappa_\perp}/\sqrt{b_x^2+b_y^2}\\ 0 & \sqrt{b_x^2+b_y^2}\sqrt{2\kappa_\perp} & 0 \end{pmatrix}\end{split}\]

Since the first column is all zeros, we only need two Weiner processes to describe the spatial diffusion. For 2D problems,

\[\begin{split}P = \frac{\sqrt{2\kappa_\perp}}{\sqrt{b_x^2+b_y^2}} \begin{pmatrix} -b_xb_z & -b_y\\ -b_yb_z & b_x \end{pmatrix}\end{split}\]

To calculate the drift velocity

\[\begin{aligned} & \boldsymbol{V}_d=\frac{cpv}{qB}\left\{\frac{1-\mu^2}{2}\frac{\boldsymbol{B}\times\nabla B}{B^2}+\mu^2\frac{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]}{B^3}+\frac{1-\mu^2}{2}\frac{\boldsymbol{B}(\boldsymbol{B}\cdot\nabla\times\boldsymbol{B})}{B^3}\right\}, \end{aligned}\]

we need

\[\begin{split}\begin{aligned} (\boldsymbol{B}\times\nabla B)_x & = B_y\partial_z B - B_z\partial_y B \\ (\boldsymbol{B}\times\nabla B)_y & = B_z\partial_x B - B_x\partial_z B \\ (\boldsymbol{B}\times\nabla B)_z & = B_x\partial_y B - B_y\partial_x B \\ \{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]\}_x & = B_y(\boldsymbol{B}\cdot\nabla)B_z - B_z(\boldsymbol{B}\cdot\nabla)B_y \\ \{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]\}_y & = B_z(\boldsymbol{B}\cdot\nabla)B_x - B_x(\boldsymbol{B}\cdot\nabla)B_z \\ \{\boldsymbol{B}\times[(\boldsymbol{B}\cdot\nabla)\boldsymbol{B}]\}_z & = B_x(\boldsymbol{B}\cdot\nabla)B_y - B_y(\boldsymbol{B}\cdot\nabla)B_x \\ \boldsymbol{B}\cdot\nabla & = B_x\partial_x + B_y\partial_y + B_z\partial_z \\ \boldsymbol{B}\cdot\nabla\times\boldsymbol{B} & = B_x (\partial_y B_z - \partial_z B_y) + \nonumber \\ & B_y (\partial_z B_x - \partial_x B_z) + B_z (\partial_x B_y - \partial_y B_x) \end{aligned}\end{split}\]

For 2D problems, these can be simplified using

\[\begin{split}\begin{aligned} (\boldsymbol{B}\times\nabla B)_x & = - B_z\partial_y B \\ (\boldsymbol{B}\times\nabla B)_y & = B_z\partial_x B \\ (\boldsymbol{B}\times\nabla B)_z & = B_x\partial_y B - B_y\partial_x B \\ \boldsymbol{B}\cdot\nabla & = B_x\partial_x + B_y\partial_y \\ \boldsymbol{B}\cdot\nabla\times\boldsymbol{B} & = B_x\partial_y B_z - B_y\partial_x B_z + B_z (\partial_x B_y - \partial_y B_x) \end{aligned}\end{split}\]