Diffusion-limited Brownian ratchet with constant load ($V=(D/a)z^2/(e^z-1-z),z=Fa/(k_{B}T)$)

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 $DF/(k_{B}T)$, the mean passage time solves $D{T}^{\prime \prime }-DF{T}^{\prime }/(k_{B}T)=-1$, $T^{\prime }(0)=0$, $T(a)=0$. It is $T(0)=a^2(e^z-1-z)/(D{z}^2)$, giving the displayed speed and zero-load limit $2D/a$. The finite-load slowdown is not simply $e^{-z}$ in this diffusion-limited regime.