Analysing diffusion limited processes in a cylinder using pair-correlation functions

Abstract

Diffusion-limited processes (DLP) are found in various physical, biological, and engineering systems, yet their quantification within complex spatial domains remains a challenge. In this study, we develop novel one-dimensional non-periodic and periodic pair-correlation functions (PCF) to assess the spatial patterns of DLP within a cylindrical domain. By refining previous PCF formulations, we introduce an efficient binning-based approach that significantly reduces computational costs, making the method feasible for large-scale simulations. Our analysis provides a comprehensive examination of PCF variability, distinguishing between global deviations from complete spatial randomness state and sampling-induced variation. An off-lattice agent-based model is implemented, successfully reproducing self-organized patterns reminiscent of classical DLP studies and aligning with fractal-like aggregation behaviours. We demonstrate the utility of periodic PCFs in capturing key spatial correlations in DLP, particularly in azimuthal and Cartesian projections, while highlighting the conditions under which non-periodic PCFs remain preferable. Our findings underscore the potential of PCFs as robust summary statistics for complex spatial models, with applications ranging from microbial colony formation and blood clotting dynamics to image analysis and classification algorithms.