Earliest diffusive target encounter among independent searchers ($\tau \simeq V/(N{k}_a)$)

If $N$ independent well-mixed searchers each encounter a small absorbing target with approximate exponential hazard $k_a/V$, their minimum waiting time has hazard $N{k}_a/V$. The Smoluchowski association rate is $k_a=4\pi D_{\mathrm{rel}}{r}_0$. Cell geometry, orientational capture and nonequilibrium spatial distributions can change this idealized estimate; an arbitrary nonexponential first-passage mean does not in general divide by $N$ exactly.