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The mean first-passage time obeys the backward equationFor $\alpha \ne 0$, direct integration givesIn particular,This increases monotonically with drift away from the target, so the constrained optimum isExpanding the exponential at zero drift yields

The adsorption mechanism is uniform along the two vertical sides, and $X$ evolves independently of $Y$. Consequently the mean first-passage time depends only on the initial $x$-coordinate. Its Kolmogorov backward equation for $a_1=1$ is

The forward partially absorbing boundary condition for a diffusion is $\mathbf{J}\cdot \mathbf{n}=\kappa p$. The boundary term in the adjoint relation isbecause the $X$ diffusion coefficient is one. It vanishes for every admissible $p$ precisely whenFor $\kappa =1$, the two Robin boundary conditions are therefore

The general solution of the ordinary differential equation isThe right condition gives $C=2$, and the left gives $D=-2/e$. At the prescribed initial position,

Each addition starts a new statistically identical waiting episode at zero gap. Thus the addition rate is the reciprocal of a mean first-passage time, not the reciprocal of the mean gap:This can also be derived without the steady density. If $T(x)$ is the mean time to reach $\delta$ starting at gap $x$, its backward Smoluchowski equation isThe first boundary is reflecting and the second absorbing. An integrating factor gives $T^{\prime }(x)=[1-e^{\beta Fx}]/(D\beta F)$, whenceAt weak force, $\tau \simeq {\delta }^2/(2D)$ is the time for a freely diffusing wall to move a monomer length, with reflection at the tip. At strong force, opening the gap requires work $F\delta$, with Boltzmann distribution suppression $e^{-\beta F\delta }$. The drift establishes a near-tip thermal length ${\ell }_F=k_{B}T/F$ and a diffusion time ${\ell }_F^2/D$; consequentlyThe exponentially long wait explains why the growth slows. There is no additional chemical waiting time in the imposed absorbing-threshold model; such a time would be needed to describe the separate reaction-limited polymerization Brownian ratchet regime.

For overdamped Brownian motion at uniform temperature in a potential energy $U$, constant diffusion coefficient $D$ gives the displayed Fokker-Planck equation, with $\beta =(k_{B}T{)}^{-1}$. The probability current is $\mathbf{J}=-D(\nabla p+\beta p\nabla U)$. Zero current recovers the Boltzmann distribution; absorbing boundaries instead permit transport and mean first-passage times. This diffusion equation should be distinguished from the similarly named coagulation equation.