current. The calculations required to check this for long domain walls are difficult quantum mechanically (for numerical reasons) but straightforward semi-classically. At the single-electron level, adiabaticity again corresponds to alignment of the spin moment with the effective field defined in Eq. (\[eqn:gradientfield\]). The results for a typical long wall (Fig. \[fig:longwalltorque\]) demonstrate that summation over all electrons produces alignment of ${\bf Q}(x)$ with ${\bf M}(x)$ so the adiabatic formula Eq. (\[eqn:adit\]) is indeed correct in this limit.

For short walls, we have carried out calculations of ${\bf N}_{\rm st}(x)$ both quantum mechanically and semi-classically. The two methods agree very well with one another (see Fig. \[fig:LLGcheck\]) but not with the proposed form Eq. (\[lt\]). Bearing in mind that, when the magnetization changes, $\hat{\bf x}'$ points along $\partial_x {\bf
M}$ and $\hat{\bf y}$ points along ${\bf M}\times\partial_x{\bf M}$, our result for the spin-transfer torque is $$\label{eqn:Tdecompose}
   {\bf N}_{\rm st}(x) = {\bf N}_{\rm ad}(x)+a(x)\hat{\bf x}' + b(x)\hat{\bf y}.$$

![Distributed spin-transfer torque for a long Nèel domain wall with $w=50$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): semi-classical calculation of ${\bf N}_{\rm st}$ (solid dots) compared to Eq. (\[eqn:adit\]) for ${\bf N}_{\rm ad}(x)$ (solid curve).[]{data-label="fig:longwalltorque"}](fig5.eps){width="8cm"}

![Distributed spin-transfer torque for a short Nèel domain wall with 