Scaling laws for molecular communication

In this paper, we investigate information-theoretic scaling laws, independent from communication strategies, for point-to-point molecular communication, where it sends/receives information-encoded molecules between nanomachines. Since the Shannon capacity for this is still an open problem, we first derive an asymptotic order in a single coordinate, i.e., i) scaling time with constant number of molecules m and ii) scaling molecules with constant time t. For a single coordinate case, we show that the asymptotic scaling is logarithmic in either coordinate, i.e., Θ(log t) and Θ(log m), respectively. We also study asymptotic behavior of scaling in both time and molecules and show that, if molecules and time are proportional to each other, then the asymptotic scaling is linear, i.e., Θ(t) = Θ(m).