Molecular Communication in H-Diffusion

Abstract

The random propagation of molecules in a fluid medium is characterized by the spontaneous diffusion law as well as the interaction between the environment and molecules. In this paper, we embody the anomalous diffusion theory for modeling and analysis in molecular communication. We employ H-diffusion to model a non-Fickian behavior of molecules in diffusive channels. H-diffusion enables us to model anomalous diffusion as the subordinate relationship between self-similar parent and directing processes and their corresponding probability density functions with two H-variates in a unified fashion. In addition, we introduce standard H-diffusion to make a bridge of normal diffusion across wellknown anomalous diffusions such as space-time fractional diffusion, Erdélyi-Kober fractional diffusion, grey Brownian motion, fractional Brownian motion, and Brownian motion. We then characterize the statistical properties of uncertainty of the random propagation time of a molecule governed by Hdiffusion laws by introducing a general class of molecular noise-called H-noise. Since H-noise can be an algebraic tailed process, we provide a concept of H-noise power using finite logarithm moments based on zero-order statistics. Finally, we develop a unifying framework for error probability analysis in a timing-based molecular communication system with a concept of signal-to-noise power ratio. ). 2 3 for timing and amplitude modulations. This work further extended to a connectivity problem with a random time constraint in a one-dimensional nanonetwork, where the random locations of molecules at the initial time are modeled by poisson point process [20]. The Cox process has been considered in [21] to capture the dynamic variation of the molecule concentration arising from the mobility of anomalously diffusive molecules, and the spatial ordering of the molecular communication performance has been characterized in terms of the error rate in the presence of interfering molecules. However, there is no comprehensive study on the modeling of anomalous diffusion channels in the context of molecular communication. Various anomalous diffusion processes typically can be modeled numerous ways including continuous random walk (CTRW), generalized diffusion equation, generalized master equation, fractional Brownian motion, and fractional kinetic equation (fractional diffusion equation) [15], [23]-[25]. 3 In particular, the CTRW simply describes diffusion of molecules in the medium with arbitrary distributions of jump lengths and waiting times. 4 In addition, the combination of a stochastic operational time-a directing process-and the self-similar parent process is equivalent to the subordination integral mechanism for the product of two random variables in the context of subordinated processes [28]-[31]. 5 This subordination law generates the solution of the fractional diffusion equation in purely analytical ways using the machinery offered by convolution properties of the Mellin transform. 6In this paper, we embody anomalous diffusion according to th...

Stochastic reaction and diffusion systems in molecular communications: Recent results and open problems

“…Instead, anomalous diffusion models are more appropriate. While there is some work in this direction [100,101,102,103,104,105], the intractability of the solutions to the resulting generalized Fokker-Planck equations introduces new difficulties for the design of detection rules.…”

Molecular Communication in Fractional Diffusive Channel

“…In a fractional diffusive medium, the analysis of Mol-Com is a challenging task. Recently, several works in the literature, including [8]- [12], have analyzed the fractional diffusive channel in MolCom systems. The first-passagetime-density (FPTD) function for the fractional diffusive medium has been obtained in [8] using the Caputo fractional derivative and the channel has been analyzed in terms of bit error rate.…”

“…The stochasticity of an anomalous diffusive channel is analyzed in [11]. Recently, authors in [12] used H-diffusion modeling to analyze the anomalous diffusion phenomenon in MolCom with a timing modulation method.…”

“…We note that in [8], [11], [12], authors have used the generalized space-time fractional diffusion equation with a composite fractional time derivative that involves the Hfunction. In [9], sub-diffusive behavior was only analyzed with the CTRW model.…”

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