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.

翻译：本文提出了一种用于分子通信（MC）系统的新型数学模型，以第一到达位置（FAP）作为信息传输的基本模式。我们解决了两个关键挑战：FAP密度的表征以及具有垂直漂移FAP通道的容量界限建立。我们的方法将宏观偏微分方程（PDE）模型与微观随机微分方程（SDE）模型相关联，得到了一个将FAP密度与椭圆型格林函数联系的精确表达式。该公式的独特之处在于其广泛适用性，可适用于任意空间维度、任意漂移方向及多种接收器几何形状。我们通过案例研究（二维和三维平面接收器）展示了模型的实际应用性，并利用基于粒子的仿真验证了公式的准确性。进一步地，明确的FAP密度形式使我们能够在二阶矩约束下建立垂直漂移FAP通道容量的闭式上界和下界，从而显著推进了对MC系统中FAP通道的理解。