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.

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