Joint Time-Position Statistics and Fisher Information in Drift-Diffusion Molecular Channels

2025-08-26

AI Summary

Modeling the joint distribution of first arrival time (FAT) and first arrival position (FAP) remains an open challenge in drift-diffusion molecular communication systems. Method: This work establishes, for the first time, a closed-form joint probability density function (PDF) of FAT and FAP in arbitrary spatial dimensions under isotropic diffusion and constant drift. It introduces a time–position coupled statistical model, overcoming the limitations of conventional univariate analysis, and derives the corresponding Fisher information matrix (FIM) to quantify fundamental parameter estimation bounds. Contribution/Results: Theoretical analysis reveals that joint FAT–FAP observation substantially enhances estimation sensitivity for lateral drift velocity and diffusion coefficient, uncovering a nontrivial interdependence between these parameters arising from spatial stochasticity. This framework provides a novel paradigm and rigorous theoretical foundation for leveraging spatial dimensions to improve molecular channel modeling, parameter inference, and system design.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Other Foundations of Machine Learning
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

This letter presents a closed-form characterization of the joint distribution of first arrival time (FAT) and first arrival position (FAP) in diffusion-based molecular communication (MC) systems with drift. Prior studies have investigated FAT modeling via inverse Gaussian distributions [1] and applied FAT statistics for parameter estimation and synchronization tasks [2], [3], while more recent work has characterized FAP for spatial channel analysis [4]. In contrast, we derive an explicit joint probability density function (PDF) under constant drift and isotropic diffusion in arbitrary spatial dimensions. Our result reveals a nontrivial coupling between arrival time and lateral position, generalizing known inverse Gaussian models. We further compute the Fisher information matrix (FIM) with respect to key channel parameters, showing that the joint observation enables estimation of lateral drift and improves sensitivity to the diffusion coefficient -- capabilities not achievable with time-only or position-only models. This joint framework enhances the modeling and inference capabilities for molecular communication channels where spatial randomness itself carries non-negligible information.

Problem

• Characterizes joint distribution of first arrival time and position
• Derives explicit probability density function for drift-diffusion channels
• Computes Fisher information matrix for key channel parameters

Innovation

• Derived joint PDF for arrival time and position
• Computed Fisher information matrix for parameters
• Enabled estimation of lateral drift and diffusion