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Knudsen Diffusion in Finite-Size Channels from a First-Passage Point of View

We studied the distribution of molecular hits on the wall of a finite cylindrical channel in the Knudsen regime. Particles entered the channel and either returned to the entrance or were transmitted to the opposite channel end. Using a first-passage approach we derived expressions for the spatial distributions of hitting probabilities. Monte Carlo simulations essentially confirmed the theoretical predictions, but significant boundary effects were observed. We related these to distributions of chord lengths {r}: g(r), characterizing chords connecting entrance positions and the locations of the first hit, and f(r), representing chords connecting two consecutive collision points. An interesting numerical observation is their asymptotic (large r) behavior: g(r) ∼ 1/r3 and f(r) ∼ 1/r4. We present analytical calculations sustaining these power laws.