= Solution
Thermal molecular motion causes the coarse droplet radius to fluctuate as well as drift down $F(R)$, so its effective equation must be an <Overdamped Langevin dynamics>. With $\langle\Lambda(t)\Lambda(t')\rangle=\delta(t-t')$, detailed balance with equilibrium density proportional to $e^{-F/(k_BT)}$ imposes the <Fluctuation-dissipation theorem>. The <Model A fluctuation-dissipation relation> gives
$$
\boxed{A=\sqrt{2k_BT\,\overline{\mathcal M}}.}
$$
The factor of mobility ensures that the diffusion in $R$ and the dissipative drift have the same equilibrium Gibbs distribution.
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