= Solution
In the <steady state>, conservation of <probability> requires $J_x=0$, so $J=j_0$ is constant. The sign is positive because increasing gap reaches the incorporation boundary, whereas the deterministic drift points towards smaller gap. Thus
$$
p'(x)+\beta Fp(x)=-\frac{j_0}{D}.
$$
An <integrating factor> $e^{\beta Fx}$ gives
$$
\boxed{p(x)=A e^{-\beta Fx}-\frac{j_0}{D\beta F}\quad(F>0).}
$$
Substitution makes the check particularly transparent:
$$
p'=-\beta F A e^{-\beta Fx},\qquad
-D(p'+\beta Fp)=j_0,
$$
so $\partial_t p=-J_x=0$. The constant offset is necessary for nonzero transport; a pure <Boltzmann distribution> would have $J=0$ and could not describe a steady succession of addition events. The $F=0$ limit is taken after applying the boundary conditions, avoiding the apparent singularity of the displayed parametrization.
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