The impulse response of the first arrival position (FAP) channel in molecular communication (MC) has been derived for spatial dimensions 2 and 3 in recent works, however, the Shannon capacity of FAP channels has yet to be determined. The fundamental obstacle to determining the capacity of FAP channels is rooted in the multi-dimensional Cauchy distribution nature of the FAP density, particularly when the drift velocity approaches zero. Consequently, conventional approaches for maximizing mutual information are inapplicable as the first and second moments of Cauchy distributions are non-existent. This paper presents a comprehensive characterization of the zero-drift FAP channel capacity for 2D and 3D spaces. The capacity formula for the FAP channel is found to have a form similar to the Gaussian channel case (under second-moment power constraint). Notably, the capacity of the 3D FAP channel is twice that of the 2D FAP channel, providing evidence that FAP channels have greater capacity as spatial dimensions increase. Our technical contributions include the application of a modified logarithmic constraint in lieu of the typical power constraint, and the selection of an output signal constraint as a replacement for the input signal constraint, resulting in a more concise formula.

翻译：近年来，已有文献推导了分子通信（MC）中首次到达位置（FAP）信道在二维和三维空间中的冲激响应，然而FAP信道的香农容量尚未确定。其根本障碍在于FAP概率密度服从多维柯西分布特性，特别是当漂移速度趋近于零时。由于柯西分布的一阶矩和二阶矩不存在，传统互信息最大化方法无法适用。本文系统描述了二维和三维空间中零漂移FAP信道容量的完整特征。研究表明，FAP信道的容量公式具有与高斯信道（在二阶矩功率约束下）相似的结构形式。值得注意的是，三维FAP信道的容量是二维FAP信道容量的两倍，这证明随着空间维度的增加FAP信道具有更大的容量。我们的技术贡献包括：用修正对数约束替代传统功率约束，以及选用输出信号约束替代输入信号约束，从而得到更简洁的公式。