= Solution
The gap is not a fixed mechanical separation. <Thermal fluctuations> of the wall compete with drift towards the <microtubule> under the opposing <force>. A <monomer> can be incorporated only after the gap reaches its length. The <Smoluchowski equation> describes this overdamped drift and <diffusion> of gap <probability density>.
For gap potential <energy> $U(x)=Fx$, the equation and its <probability current> are
$$
\partial_t p=-\partial_xJ,
\qquad J=-D\left(\partial_xp+\beta U'(x)p\right)
=-D(p_x+\beta Fp),\qquad \beta=(k_BT)^{-1}.
$$
The drift mobility is $D\beta$, the overdamped consequence of the <fluctuation-dissipation relation for a Langevin particle>. Equivalently, this coefficient is fixed by requiring the zero-current solution to be the <Boltzmann distribution> $p\propto e^{-\beta U}$. Solve this equation with the appropriate exclusion, incorporation and probability-normalization conditions. The steady flux through the incorporation threshold gives the addition rate $j_0$; the elongation speed is $v=\delta j_0$.
There is a regime distinction in the printed problem. The absorbing boundary imposed in part (c) means instantaneous incorporation once a gap opens, and leads to the <diffusion-limited Brownian ratchet with constant load>. A genuinely <reaction-limited polymerization Brownian ratchet> instead assumes fast gap equilibration followed by a finite chemical addition rate, giving an exponential gating factor multiplying that chemical rate. The equations requested in the subsequent parts specify the former model, which is the one solved here.
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