current density[@Qnote] ${\bf Q}_\pm(x,k_x)$ for an up/down ($\pm$) electron with longitudinal wave vector $k_x$.

Summing over all electrons and using the relaxation time approximation gives the non-equilibrium majority and minority spin density ${\bf
s}_\pm(x)$ and spin current density ${\bf Q}_\pm(x)$ in the presence of an electric field $E\hat{\bf x}$: $$\begin{aligned}
\label{eqn:spinandcurrent}
{\bf s}_\pm(x) 
    &=& \int{\left[ f_\pm({\bf k}-{eE\tau\over\hbar}\bm{\hat{x}})-f_\pm({\bf k}) \right]}{\bf s}_\pm(x,k_x)d^3{\bf k} \nonumber \\
& & \\
{\bf Q}_\pm(x) 
    &=& \int{\left[ f_\pm({\bf k}-{eE\tau\over\hbar}\bm{\hat{x}})-f_\pm({\bf k}) \right]}{\bf Q}_\pm(x,k_x)d^3{\bf k} \nonumber\end{aligned}$$ Our use of the function $f_\pm({\bf k})=\Theta(k_F^\pm-{\left| \bf k \right|})$ implies that the distribution of electrons outside the region of inhomogeneous magnetization are characteristic of the zero-temperature bulk.[@linearBoltzmann] We shall expand this point and comment on the general correctness of (\[eqn:spinandcurrent\]) in Sec. \[scattering\]. The sum ${\bf s}(x)={\bf s}_+(x)+{\bf s}_-(x)$ is the total spin accumulation (spin density) and ${\bf Q}(x)={\bf Q}_+(x)+{\bf Q}_-(x)$ is the total spin current density. Finally, the distributed spin transfer torque is[@Stiles:2002] $$\label{eqn:spt}
{\bf N}_{\rm st}(x)=-\partial_x{\bf Q}(x).$$ The adiabatic approximation[@Berger:1978] to the spin dynamics leads to a spin current density that is proportional to the local magnetization, ${\bf Q}_{\rm ad}(x) \propto {\bf M}(x)$. This means that $$\label{eqn:adia}
{\bf N}_{\rm ad}(x)  \propto \partial_x {\bf M}(x).$$ A main goal of this paper 