  This paper introduces a novel mathematical model for Molecular Communication
(MC) systems, utilizing First Arrival Position (FAP) as a fundamental mode of
information transmission. We address two critical challenges: the
characterization of FAP density and the establishment of capacity bounds for
channels with vertically-drifted FAP. Our method relate macroscopic Partial
Differential Equation (PDE) models to microscopic Stochastic Differential
Equation (SDE) models, resulting in a precise expression that links FAP density
with elliptic-type Green's function. This formula is distinguished by its wide
applicability across any spatial dimensions, any drift directions, and various
receiver geometries. We demonstrate the practicality of our model through case
studies: 2D and 3D planar receivers. The accuracy of our formula is also
validated by particle-based simulations. Advancing further, the explicit FAP
density forms enable us to establish closed-form upper and lower bounds for the
capacity of vertically-drifted FAP channels under a second-moment constraint,
significantly advancing the understanding of FAP channels in MC systems.
