Mathematical Modeling of Impurity Diffusion Processes in a Multiphase Randomly Inhomogeneous Body Using Feynman Diagrams

• Published in: Symmetry (Basel) Vol. 17; no. 6; p. 920
• Main Authors: Pukach, Petro, Chernukha, Yurii, Chernukha, Olha, Bilushchak, Yurii, Vovk, Myroslava
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 10.06.2025
• Subjects: Algorithms; Analysis; Body volume (biology); Cognition & reasoning; Configuration management; Differential equations; Diffusion layers; Feynman diagrams; Feynman, Richard P; Impurities; Inhomogeneous media; Mass transfer; Modelling; Multiphase; Operators (mathematics); Parabolic differential equations; Partial differential equations; Phase distribution; Physics; Quantum field theory; Symmetry; Theoretical physics; Velocity; Iran; Germany; Ukraine
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym17060920

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume.