Characterizing First Arrival Position Channels: Noise Distribution and Capacity Analysis

TL;DR

This work develops a rigorous framework for molecular communication channels that use First Arrival Position (FAP) as the information carrier. By linking macroscopic diffusion PDEs with microscopic Itô diffusion via the generator and elliptic Green’s functions, it derives a unified expression for the FAP density applicable to arbitrary dimensions, drift directions, and receiver geometries. The authors then introduce vertically-drifted FAP (VDFAP) distributions, analyze their frequency-domain characteristics, and establish closed-form lower and upper bounds on channel capacity under a second-moment constraint, with verification through particle-based simulations. The results offer a principled way to quantify and optimize information transfer in MC systems, including potential applications to Molecular MIMO and Molecular IM. The methodology also provides a pathway to handle diverse receiver shapes and drift scenarios in future MC designs.

Abstract

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.

Figures (5)

• Figure 1: This figure illustrates a 2D FAP channel with line-shaped Tx and Rx, where the Tx is assumed to be transparent, allowing particles to move through it without experiencing any force. The emission point is located on the Tx line. Note that the two red particles are highlighted as exemplary particles to indicate the emission position $(x_1, x_2)$ and the hitting position $(\xi, \eta)$, respectively.
• Figure 2: This figure illustrates a 3D FAP channel with plane-shaped Tx and Rx, where the Tx is assumed to be transparent, allowing particles to move through it without experiencing any force. The emission point is located on the Tx plane. Note that the two red particles are highlighted as exemplary particles to indicate the current position $(x_1, x_2, x_3)$ and the hitting position $(\xi, \eta, \zeta)$, respectively.
• Figure 3: Comparison of particle-based simulated and theoretical FAP densities on a planar receiver (i.e., $d=2$) for zero drift $\mathbf{u}=[0,0,0]$.
• Figure 4: Comparison of particle-based simulated and theoretical FAP densities on a planar receiver (i.e., $d=2$) for a case of non-zero normalized drift $\mathbf{u}=[2,-3,-1]~(\mu\mathrm{m}^{-1})$.
• Figure 5: Numerical evaluations of derived lower bound \ref{['eq:lower-bound']} and upper bound \ref{['eq:upper-bound']} on the capacity of VDFAP channels when $d=2$, illustrating the influence of $P$, $\lambda$, and $u$ on the trends of capacity.

Theorems & Definitions (10)

• Lemma : Weak Stability Property for VDFAP
• proof
• Theorem 1: Lower Bound on $C$
• proof
• Corollary 1: Lower Bound on $C$ for $d=2$
• proof
• Theorem 2: Upper Bound on $C$
• proof
• Corollary 2: Upper Bound on $C$ for $d=2$
• proof