Mathematical Modeling of Impurity Diffusion Processes in a Multiphase Randomly Inhomogeneous Medium Under the Action of Internal Mass Sources: Feynman Diagrams Approach

Petro Pukach (), Yurii Chernukha, Olha Chernukha and Myroslava Vovk
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Petro Pukach: Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Yurii Chernukha: Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Olha Chernukha: Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Myroslava Vovk: Department of Mathematics, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine

Mathematics, 2025, vol. 13, issue 21, 1-36

Abstract: We develop a new mathematical framework for describing impurity diffusion in multiphase, stochastically inhomogeneous media with internal deterministic mass sources. The main contribution of the paper is the structural preservation of the original multiphase problem while reducing it to a single integro-differential diffusion equation for the entire body. Using a Feynman diagram technique, we obtain a Dyson-type equation for the averaged concentration field; its kernel (mass operator) summarizes the cumulative effect of random phase interfaces and internal sources. This diagrammatic formulation offers clear advantages: it systematically organizes the contributions of complex interphase interactions and source terms, ensures convergence of the Neumann-series solution, and facilitates extensions to more intricate source distributions. The approach allows us to analyze the behavior of the averaged impurity concentration under various temporally or spatially distributed internal sources and provides a foundation for further refinement of transport models in complex multiphase systems.

Keywords: diffusion; multiphase randomly inhomogeneous medium; mass source; non-ideal contact conditions; Neumann series; averaging over an ensemble of phase configurations; Feynman diagram; mass operator kernel; Dyson equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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