Optimal probe density for Brownian first-passage detection

Abstract

We simulate a two-dimensional Brownian motion of polystyrene probe particles encountering a smaller foreign target using LAMMPS to study how probe density affects the mean first-passage detection time ⟨T⟩. As the number of probes N_p increases, ⟨T⟩ first decreases in a diffusion-limited regime before increasing in a crowding-limited regime beyond ϕ ≈ 41–45%, where excluded-volume interactions dominate. The ergodic product E = N_p × ⟨T⟩ identifies this transition earlier and more sensitively than ⟨T⟩ alone, thus establishing an optimal probe density for maximizing the efficiency of foreign particle detection.