Molecular communication emerges as a promising communication paradigm for nanotechnology. However, solid mathematical foundations for information-theoretic analysis of molecular communication have not yet been built. In particular, no one has ever proven that the channel coding theorem applies to molecular communication, and no relationship between information rate capacity (maximum mutual information) and code rate capacity (supremum achievable code rate) has been established. In this paper, we focus on a major sub-class of molecular communication -the diffusion-based molecular communication. We provide solid mathematical foundations for information theory in diffusion-based molecular communication by creating a general diffusionbased molecular channel model in measure-theoretic form and prove its channel coding theorems. Various equivalence relationships between statistical and operational definitions of channel capacity are also established, including the most classic information rate capacity and code rate capacity. As byproducts, we have shown that the diffusion-based molecular channel is with "asymptotically decreasing input memory and anticipation" and " d-continuous". Other properties of diffusion-based molecular channel such as stationarity or ergodicity are also proven.
Index TermsMolecular communication, diffusion process, diffusion-based molecular system, channel capacity, nanotechnology, d-continuous channel, asymptotically decreasing input memory and anticipation channel, permutation channel, cascade of channels.
I. INTRODUCTIONM OLECULAR communication is a recently developed communication paradigm whose communication process involves "transmission and reception of information encoded in molecules" [1]. It is born in the study of nano-scale communication in nanonetworks [1] where the applicability of classical electromagnetic communication is limited by several constraints and a novel solution is called for. Inspired by the biological communication process such as the intra-and inter-cellular communication by cells [2] and the pheromone diffusion by insects [3], the researchers have created various message-carrying molecules and their corresponding receptors. Since the molecules are themselves at the nano-scale, the difficulty of nano-scale communication is solved by building up communication media between the transmitter and the receptor. Moreover, since the major potential application of nanonetworks is implemented in the organisms body (e.g., organ monitoring in human body), the compatibility of the bioinspired communication process makes it a perfect choice to consider the encoding, transmission, and decoding of messages in molecules. Excited by the reasons above (and beyond), the researchers have treated the molecular communication as one of the mainstream subjects of nanotechnology nowadays.Among the various proposed system blue maps for molecular communication, a specific class of system called the "diffusionbased molecular system" draws lots of interest due to the universality of mass transp...