Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

• Published in: Mathematics (Basel) Vol. 10; no. 18; p. 3235
• Main Authors: Cleland, Josiah D., Williams, Martin A. K.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2022
• Subjects: Asymptotic properties; Diffusion; Exact solutions; Fokker-Planck equation; Fokker–Planck; fractional; generalized; Heat equation; Laplace transforms; Mathematical analysis; Mathematical research; Probability density functions; Random walk; Random walks (Mathematics); Strains and stresses; Stress relaxation (Materials); Stress relieving (Materials); New Zealand
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10183235

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations.