Characterizing First Arrival Position Channels: Noise Distribution and Capacity Analysis

This paper addresses two fundamental problems in diffusive molecular communication: characterizing the first arrival position (FAP) density and bounding the information transmission capacity of FAP channels. Previous studies on FAP channel models, mostly captured by the density function of noise, have been limited to specific spatial dimensions, drift directions, and receiver geometries. In response, we propose a unified solution for identifying the FAP density in molecular communication systems with fully-absorbing receivers. Leveraging stochastic analysis tools, we derive a concise expression with universal applicability, covering any spatial dimension, drift direction, and receiver shape. We demonstrate that several existing FAP density formulas are special cases of this innovative expression. Concurrently, we establish explicit upper and lower bounds on the capacity of three-dimensional, vertically-drifted FAP channels, drawing inspiration from vector Gaussian interference channels. In the course of deriving these bounds, we unravel an explicit analytical expression for the characteristic function of vertically-drifted FAP noise distributions, providing a more compact characterization compared to the density function. Notably, this expression sheds light on a previously undiscovered weak stability property intrinsic to vertically-drifted FAP noise distributions.