Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model - Chattopadhyay, Amit K.; Kundu, Bidisha; Nath, Sujit Kumar; Aifantis, Elias C. (Frontiers in Applied Mathematics and Statistics, 2022)
Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals [1, 2, 3] and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model [4, 5, 6, 7, 8, 9]. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction number R_0, which is the average number of secondary infections generated from every infected. We show that the R_0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model.