Analytical derivation of the impulse response for the bounded 2-D diffusion channel

This letter focuses on the derivation of the hitting probabilities of diffusing particles absorbed by an agent in a bounded environment. In particular, we analogously consider the impulse response of a molecular communication channel in a 2-D and 3-D environment. In 2-D, the channel involves a point transmitter that releases molecules to a circular absorbing receiver that absorbs incoming molecules in an environment surrounded by a circular reflecting boundary. Considering this setup, the joint distribution of the molecules on the circular absorbing receiver with respect to time and angle is derived. Using this distribution, the channel characteristics are examined. Then, we extend this channel model to 3-D using a cylindrical receiver and investigate the channel properties. We also propose how to obtain an analytic estimate for the unbounded 2-D channel from our derived solutions, as no analytic derivation for this channel is present in the literature. Throughout the letter, we perform particle-based simulations to compare the analytic results and lay evidence for our findings.