= Diffusion-limited Brownian ratchet with constant load
{title2=$V=(D/a)z^2/(e^z-1-z),\quad z=Fa/(k_BT)$}
In the instantaneous-addition, negligible-removal ratchet, the gap reflects at zero and a first passage to $a$ triggers addition and resets the gap to zero. With load diffusion coefficient $D$ and opposing drift $DF/(k_BT)$, the mean passage time solves $DT''-DFT'/(k_BT)=-1$, $T'(0)=0$, $T(a)=0$. It is $T(0)=a^2(e^z-1-z)/(Dz^2)$, giving the displayed speed and zero-load limit $2D/a$. The finite-load slowdown is not simply $e^{-z}$ in this diffusion-limited regime.
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