= Solution
<velocity> $U$, <length> $\ell$ and <diffusivity> $D$ have dimensions $L/T$, $L$, and $L^2/T$. Their dimensionless <Péclet number> is
$$
\boxed{\mathrm{Pe}=\frac{U\ell}{D}=\frac{\ell^2/D}{\ell/U}.}
$$
<diffusion> dominates for $\mathrm{Pe}\ll1$, <advection> for $\mathrm{Pe}\gg1$. At fixed $U,D$, the crossover <length> is $D/U$; neglecting <advection> requires $\ell\ll D/U$.
TODO: user-deferred organism-specific swimming-speed and detection-scale estimates. No biological numerical substitution is made.
For the generic <equilibrium> transport derivation, let an external potential $V(x)$ give <force> $F=-V'$ and drift $u=F/\zeta$. Boltzmann <equilibrium> has $c\propto e^{-V/(k_BT)}$, hence $c'=Fc/(k_BT)$. Substituting in the zero-current condition gives
$$
J=Fc\left(\frac1\zeta-\frac{D}{k_BT}\right)=0,
\qquad\boxed{D=\frac{k_BT}{\zeta}=\frac{k_BT}{6\pi\eta a}.}
$$
The <Stokes–Einstein relation> in the final equality is for an isolated no-slip sphere in a Newtonian liquid at low <Reynolds number>. Zero flux alone is insufficient: the <equilibrium> distribution and mechanical mobility law are also needed.
TODO: user-deferred organism-specific absorbing-surface calculation and its biological flux interpretation.
TODO: user-deferred receptor-number, receptor-density and receptor-competition calculations. These continuations have not been solved.
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